LAWS  OF 
PHYSICAL  SCIEtCE 

EDWIN  F.NORTHRUP,PH.D. 


LAWS  OF 
PHYSICAL  SCIENCE 

A  REFERENCE  BOOK 


BY 

EDWIN  F.  NORTHRUP,  PH.D. 

PALMER  PHYSICAL  LABORATORY,  PRINCETON  UNIVERSITY,  PRINCETON,  N.  J. 


PHILADELPHIA  AND  LONDON 
J.  B.  LIPPINCOTT  COMPANY 


COPYRIGHT,  1917 
BY  J.   B.   LIPPINCOTT  COMPANY 


Electrotyped  and  Printed  by  J.  B.  Lippincott  Company. 
The  Washington  Square  Press,  Philadelphia,  U.  S.  A. 


THIS  COLLECTION  OF  NATURE'S  LAWS  IS 
LOVINGLY  DEDICATED  TO  AN  ABLE  AND  MOST 
HONORABLE  EXPONENT  OF  HUMAN  LAWS 

MY  FATHER 
HON.  ANSEL  JUDD  NORTHRUP 


3G0446 


PREFACE 

EXACT  knowledge  consists  of  accumulated  facts  and  sets 
of  formulated  propositions  respecting  facts.  Data,  Mathe- 
matical relations  and  Physical  laws  constitute  the  three 
firm  supports  of  Physical  science  and  Engineering. 

The  data  of  physical  science  are  readily  accessible  in 
several  published  tables  of  physical  constants.  The  mathe- 
matics used  in  physical  science  has  been  summarized,  classi- 
fied and  formulated,  for  ready  reference,  in  many  published 
books.  The  author  is  not  aware,  however,  of  any  hand-book 
or  reference  work  which  contains  a  full  list  of  the  general 
propositions  or  laws  of  science. 

Such  reference  lists  are  not  without  value,  and  this  book 
has  been  prepared  to  fill  an  obvious  gap  in  the  literature  of 
Physical  Science.  Furthermore,  it  appears  to  the  author 
that  students  in  any  of  the  branches  of  Natural  Science  will 
not  only  find  guidance,  but  will  also  derive  inspiration  by 
having  before  them  under  a  single  view  the  very  epitome  of 
the  world's  heritage  of  the  fundamentals  of  its  knowledge 
and  wisdom.  None  will  question  that  the  fundamentals  of 
science  are  its  laws,  principles,  theorems  and  precise  state- 
ments of  the  general  properties  of  matter;  but  it  is  not 
always  easy  for  students  in  one  branch  of  science  to  find  and 
to  know  the  literature  on  important  principles  and  facts  in 
an  entirely  different,  or  even*  in  closely  allied  branches  of 
science.  The  author  hopes  that  what  has  been  here  gath- 
ered together  and  classified  will  help  such  students  in  their 
search  and  give  them  the  means  to  broaden  their  view. 

We  have  chosen  for  a  title,  * '  Laws  of  Physical  Science  ' ' 
but  many  general  propositions,  theorems  and  mere  state- 
ments of  important  facts  have  been  included  which  perhaps, 


vi  PREFACE 

if  strictly  considered,  could  not  be  discriminated  as  laws. 
Indeed,  it  was  found  impossible,  in  many  cases,  to  decide  if 
certain  propositions  possess  sufficient  generality  and  validity 
to  deserve  the  title  "law."  When,  however,  such  doubts 
existed,  a  policy  of  inclusion  has  been  followed  in  preference 
to  one  of  exclusion. 

For  convenience  and  system  the  general  statements .  (in 
all  480  with  title)  have  been  classified  in  six  sections :  I — Me- 
chanics ;  II — Hydrostatics,  Hydrodynamics  and  Capillarity ; 
III— Sound ;  IV— Heat  and  Physical  Chemistry ;  V^  -Elec- 
tricity and  Magnetism ;  VI — Light. 

Each  law,  proposition  or  general  statement  is  charac- 
terized by  giving  it  a  heading  or  title.  Each  proposition 
covered  by  a  title  is  followed  by  one,  and  in  many  .cases  by 
several  references  to  easily  accessible  text-books,  standard 
treatises,  and,  in  a  few  cases,  to  original  articles  or  papers, 
where  one  may  find  the  propositions  stated  in  different  forms 
and  additional  information  concerning  them  of  authorita- 
tive character. 

While  many  laws  of  Physical  Science  have  had  their 
origin  with  individual  investigators,  the  perfected  form  of 
statement  they  now  possess  has  been  in  the  main  reached 
by  a  process  of  intellectual  growth  in  which  many  have 
taken  part.  It  has  seemed,  therefore,  wiser  to  make  most  of 
the  references  to  treatises  and  text-books  on  physics,  physi- 
cal chemistry  and  chemistry,  rather  than  to  papers  written 
by  the  authors  of  the  propositions.  Moreover,  original 
papers,  beside  containing  much  extraneous  matter,  are  not 
usually  readily  accessible  as  are  text-books  and  treatises. 

An  alphabetically  arranged  bibliography  of  all  books  and 
journals  referred  to  and  a  very  full  index,  with  duplicated 
references,  to  aid  in  the  quick  location  of  subject  matter  and 
proper  names,  are  included. 


PREFACE  vii 

The  author  expresses  his  acknowledgment  to  Mr.  H.  A. 
'Frederick  for  assistance  in  collecting  some  of  the  material 
used,  and  to  Prof.  K.  T.  Compton  for  his  careful  reading  of 
the  manuscript.  He  further  acknowledges  with  gratitude 
the  unfailing  and  valuable  assistance  of  his  wife,  Margaret 
Stewart  Northrup,  in  collecting  material  and  in  arranging 
it  for  publication. 

PALMER  PHYSICAL  LABORATORY  THE  AUTHOR 

PRINCETON,  N.  J. 
JANUARY,  1917. 


CONTENTS 

PASS 

I.  MECHANICS 1 

II.  HYDROSTATICS,  HYDRODYNAMICS  AND  CAPILLARITY 29 

III.  SOUND 43 

IV.  HEAT  AND  PHYSICAL  CHEMISTRY 59 

V.  ELECTRICITY  AND  MAGNETISM Ill 

VI.    LIGHT , 163 

BIBLIOGRAPHY  AND  INDEX .  191 


I 

MECHANICS 


LAWS  OF  PHYSICAL  SCIENCE 
MECHANICS 

NEWTON'S   FIRST   LAW   OF   MOTION. 

Every  body  continues  in  its  state  of  rest  or  of  uniform 
motion  in  a  straight  line,  except  in  so  far  as  it  may  be  com- 
pelled by  force  to  change  that  state. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  I,  art.  244.) 

NEWTON'S  SECOND  LAW  OF  MOTION. 

Change  of  motion  is  proportional  to  force  applied,  and 
takes  place  in  the  direction  of  the  straight  line  in  which  the 
force  acts. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  I,  art.  251.) 

NEWTON'S   THIRD    LAW   OF   MOTION. 

To  every  action  there  is  always  an  equal  and  contrary 
reaction ;  or,  the  mutual  actions  of  any  two  bodies  are  always 
equal  and  oppositely  directed. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  I,  art.  261.) 


4  LAWS  OF  PHYSICAL  SCIENCE 

NEWTON'S  LAW  OF  UNIVERSAL  GRAVITATION. 

All  bodies  attract  each  other  with  a  force  proportional  to 
the  product  of  their  masses  and  inversely  proportional  to 
the  square  of  the  distance  between  them. 


Thus,  F  =  G 


MM' 


j.2 

G  is  called  the  Newtonian  constant. 

Cavendish  (1797-1798)  first  measured  G  experimentally. 
In  1874  C.  V.  Boys  obtained  6.6576  X  10-8  on  the  C.G.S. 
system  for  the  value  of  G. 

(Consult  Scientific  Memoirs,  edited  by  J.  S.  Ames,  Vol. 
IX,  The  Laws  of  Gravitation.  See  p.  136.) 

NEWTON'S  LAW  OF  ATTRACTION  FOR  A  SPHERE. 

The  gravitational  attraction  of  a  particle  toward  a 
sphere,  whose  density  may  be  a  function  of  the  distance  from 
the  center,  is  the  same  as  if  the  mass  of  sphere  were  concen- 
trated in  a  particle  at  the  center  of  the  sphere. 

The  attraction  by  such  a  sphere  on  a  particle  within  it  is 
due  entirely  to  that  portion  of  the  sphere  which  lies  inside 
of  a  concentric  surface  through  the  particle.  In  other  words, 
the  attractions  due  to  all  parts  of  the  sphere  which  lie 
farther  than  the  particle  from  the  center  cancel  each  other. 

(Consult  Thomson  and  Tait,  Treatise  on  Natural 
Philosophy,  Part  II,  arts.  462,  471  et  seq.) 

WEIGHT    AND    MASS. 

Bodies  are  constant  in  mass  but  variable  in  weight,  and 
weight  is  always  proportional  to  mass. 

Thus,  W  =  Mg, 

where  W  is  the  weight  of  the  body,  M  its  mass  and  g  the 
acceleration  of  gravity. 

The  value  of  g  increases  from  the  equator  to  the  poles, 
g  =  978.00  cm.  per  sec.  per  sec.  at  the  equator  and  g  = 
983.01  at  the  poles. 


MECHANICS  5 

(Jeans,  Theoretical  Mechanics,  pp.  29-31.  For  values 
of  g  at  various  places  consult  Chwolson,  Traite  de  Physique, 
Vol.  I,  Part  3,  p.  387.  For  accepted  mean  value  of  g  consult 
.  Hering,  Conversion  Tables,  p.  87.) 

LAW   OF   FALLING  BODIES. 

The  law  by  which  bodies  fall  in  vacuum  is :  The  velocity 
of  descent  is  proportional  to  the  time  of  falling,  and  the 
distance  of  descent  is  proportional  to  the  square  of  the  time 
of  falling. 

Thus,  v  co  t,  and  s  oc  t2. 

Or,  v  =  gt  and  s=  fjl  when  the  body  starts  from  rest. 
Or,  d—  —  S)  which  integrated  gives, 

s  =  ^  gt2  +  Ai  t  +  A,. 

Here  g  is  the  constant  acceleration  of  gravity.    Aj  and  A2 
are  constants  of  integration. 

(Consult  Mach,  Science  of  Mechanics,  p.  130  et  seq.  See 
''Use  of  Analogy  in  Viewing  Physical  Phenomena"  by 
E.  F.  Northrup,  Jour.  Frank.  Inst.,  p.  17,  July,  1908.) 

DESCENT    ON   AN   INCLINED    PLANE. 

The  acceleration  along  an  inclined  plane  AC  is  to  the 
acceleration  along  the  vertical  AB  as  the  length  AB  is  to 
the  length  AC,  or, 

acceleration  along  inclined  plane  AB    _  «rii/n 

— : ". —  r~~~          ~  —   — .  _.    —  sin  a.  wiiere  a 

acceleration  or  gravity  AC 

=  the  angle  of  inclination  of  the  plane. 

(Consult  Mach,  Science  of  Mechanics,  pp.  137-138.) 


«  LAWS  OF  PHYSICAL  SCIENCE 

CONDITION  OF  EQUILIBRIUM  FOR  A  SYSTEM. 

When  a  system  of  particles  is  in  equilibrium  under  the 
action  of  any  system  of  external  forces,  the  sum  of  the 
components  of  all  these  forces  in  any  direction  is  zero ;  and 
the  sum  of  the  moments  of  all  these  forces  about  any  line 
is  zero. 

(Consult  Jeans,  Theoretical  Mechanics,  p.  64.  For  com- 
plete treatment,  see  Thomson  and  Tait,  Treatise  on  Natural 
Philosophy,  Part  II,  Chaps.  VI  and  VII.) 

BASIC  EQUATIONS  OF  MECHANICS;  COMMENT  ON. 

Let  s  =  distance,  t  =  time,  v  =  instantaneous  velocity, 
a  =  acceleration  of  a  uniformly  accelerated  motion,  F  = 
ma  =  force  and  m  =  mass.  Assuming  the  body  starts  from 
rest  the  relations  of  these  quantities  may  be  placed  in  the 
two  groups ; 

v  =  at      )  mv=Ft       ) 

s=^atni  ms  =  HFt2  [II 

as  =  Hv2  J  Fs  =  ^mv2) 

Equations  of  group  I  contain  the  quantity  a,  and  in  addi- 
tion two  of  the  quantities  s,  t,  v,  as  in  table  A,  and  equations 
of  group  II  contain  the  quantities  m,  F,  s,  t,  v,  each  equation 
containing  m  F  and  in  addition  to  m  F  two  of  the  quan- 
tities s,  t,  v,  as  in  table  B, 


f  v,t  f  v, 

(A)    a   1    s,  t  (B)    m,  P  1   s, 

(     8,  V  IS, 

(Mach,  Science  of  Mechanics,  pp.  269-270.) 


MECHANICS  7 

GENERAL    MECHANICAL   PRINCIPLE. 

"  A  system  always  tends  to  move  from  rest  in  such  a 
way  as  to  diminish  the  potential  energy  as  much  as  possible, 
and  the  force  tending  to  assist  a  displacement  in  any  direc- 
tion is  equal  to  the  rate  of  diminution  of  the  potential 
energy  in  that  direction." 

( J.  J.  Thomson,  Elements  of  Electricity  and  Magnetism, 
p.  82.) 

KEPLER'S   FIRST   LAW. 

The  planets  move  about  the  sun  in  ellipses,  at  one  focus 
of  which  the  sun  is  situated. 

(Consult  Mach,  Science  of  Mechanics,  p.  187.) 

KEPLER'S   SECOND  LAW. 

The  radius  vector  joining  each  planet  with  the  sun  de- 
scribes equal  areas  in  equal  times. 

(This  second  law,  the  law  of  areas,  can  be  explained 
simply  if  it  be  assumed,  as  a  particular  case,  that  the 
acceleration  toward  the  sun  is  constant.) 

(Consult  Mach,  Science  of  Mechanics,  p.  188.) 

KEPLER'S  THIRD  LAW. 

The  cubes  of  the  mean  distances  of  the  planets  from  the 
sun  are  proportional  to  the  squares  of  their  times  of  revolu- 
tion about  the  sun.  This  law  may  be  stated  mathematically, 

R;    R;     R; 

'^==^T=jr=----=a  constant 

Here  R1?  R2>  &3>  are  mean  radii  and  T±>  T2,  T3,  etc.,  the 
respective  times  of  revolution  of  the  planets. 

(Consult  Mach,  Science  of  Mechanics,  pp.  188-189. 
Also  Laplace,  Traite  de  Mecanique  Celeste,  Book  II, 
Chap.  I.) 


8  LAWS  OF  PHYSICAL  SCIENCE 

LAW  OF  THE  LEVER. 

Any  lever  is  in  equilibrium  when  the  algebraic  sum  of 
the  statical  moments  taken  about  the  fulcrum  equals  zero. 

By  the  statical  moment  is  here  meant  the  force  acting  at 
a  point  multiplied  by  the  perpendicular  distance  from  the 
line  of  support  to  the  direction  of  the  force. 

(For  a  lucid  discussion  of  the  principle  of  the  lever  see 
Mach,  Science  of  Mechanics,  Chap,  I.) 

COMPOSITION    BY    PARALLELOGRAM    RULE. 

If  a  parallelogram  A  B  C  D  is  so  drawn  that  two  of  its 
sides,  as  A  B,  A  D,  which  meet  in  a  point,  represent  as 
regards  both  magnitude  and  direction,  two  displacements,  or 
two  velocities,  or  two  accelerations,  or  two  forces  to  be  com- 
pounded, then  the  diagonal  A  C  of  this  parallelogram  will 
represent,  in  respect  to  both  magnitude  and  direction,  the 
resultant  displacement,  velocity,  acceleration  or  force. 

(Consult  Jeans,  Theoretical  Mechanics,  Chaps.  I  and 

in.) 

PARALLELOGRAM    OF   FORCES. 

Two  forces  acting  at  the  same  time  and  for  the  same 
time  upon  a  particle  produce  accelerations  which  are  in- 
dependent of  each  other  and  proportional  to  the  forces. 
The  resultant  distance,  direction,  velocity  or  acceleration 
is  obtained  by  compounding  by  the  parallelogram  rule. 

(Consult  Jeans,  Theoretical  Mechanics,  Chap.  III.) 

LAMI'S    THEOREM. 

"  When  a  particle  is  acted  on  by  three  forces,  the 
necessary  and  sufficient  condition  for  equilibrium  is  that 
the  three  forces  shall  be  in  one  plane  and  that  each  force 
shall  be  proportional  to  the  sine  of  the  angle  between  the 
other  two." 

(For  proof  see  Jeans,  Theoretical  Mechanics,  p.  40.) 


MECHANICS  0 

CENTRIPETAL  ACCELERATION. 

The  centripetal  acceleration  of  a  body  moving  in  a  circle 
is  directed  toward  the  center  of  the  circle  and  is  equal  to  the 
square  of  its  instantaneous  linear  velocity  divided  by  its 
distance  from  the  center  about  which  it  rotates  at  the  instant 
considered. 
For  uniform  rotation  in  a  circle, 


acceleration  =    velocity       ,  or,  a  =  -*- . 
radius  r 

(Consult  Laplace,  Traite  de  Mecamque  Celeste,  Book  I, 
art.  10.) 

PRINCIPLES  OF  THE  SCREW  AND   THE  WRENCH. 

Any  motion  can  be  reduced  to  a  translation  and  a  rota- 
tion about  an  axis  parallel  to  the  translation.  Such  a  com- 
bination of  a  translation  and  a  rotation  is  called  a  Screw. 

Similarly  any  system  of  forces  acting  on  a  rigid  body 
may  be  replaced  by  a  single  force  and  a  couple  about  the 
line  of  the  force.  This  combination  of  a  force  and  a  couple 
is  known  as  a  Wrench. 

Thus  a  screw  and  a  wrench  are  the  most  general  types 
of  motion  and  force  respectively. 

(Jeans,  Theoretical  Mechanics,  pp.  91,  106-107.  For 
general  treatment  of  motion  in  three  dimensions,  see  Routh, 
Elementary  Rigid  Dynamics,  Chap.  V,  pp.  184-228.) 


10  LAWS  OF  PHYSICAL  SCIENCE 

THEOREM    OF  PRINCIPAL  AXES   OF  INERTIA. 

In  every  rigid  body  there  are  three  mutually  perpen- 
dicular axes  of  rotation  intersecting  at  the  center  of  gravity 
of  the  body  which  are  characterized  by  the  property  that, 
if  the  body  be  rotated  about  any  one  of  these  axes,  it  will 
continue  thus  to  rotate  unless  acted  on  by  an  external  force. 
If  rotating  about  any  other  axis  the  centrifugal  forces  tend 
to  change  the  axis  of  rotation  and  the  rotation  is  therefore 
unstable.  The  position  of  these  axes  is  determined  mathe- 
matically by  the  fact  that,  if  they  be  taken  as  the  axes  of 
coordinates,  the  products  of  inertia. 

JTmxy,  Jfmyx,  J'mzx 

for  the  body  will  vanish. 

(Webster,  The  Dynamics  of  Particles  and  of  Rigid, 
Elastic  and  Fluid  Bodies,  pp.  228-229.  Also  Routh,  Ele- 
mentary Rigid  Dynamics,  pp  12-13.) 

MOMENT   OF  INERTIA  AND  ENERGY  OF  ROTATION. 

The  moment  of  inertia  about  any  straight  line  through 
a  rigid  body  made  up  of  a  system  of  particles  is  the  sum 
of  the  products  of  the  mass  of  each  particle  by  the  square 
of  its  perpendicular  distance  from  this  straight  line;  its 
axis  of  rotation. 

Thus,  moment  of  inertia  =  I  =  .Jmr2  where  m  is  the 
mass  of  a  particle  and  r  its  perpendicular  distance  from  its 
axis  of  rotation. 

The  kinetic  energy  Er  of  the  rotating  body  is,  Er  = 
%  I  w2,  where  o>  is  its  angular  velocity.  This  is  the  analogue 
of  Ej  =  %  mv2,  where  m  is  the  mass  of  a  body  having  a 
rectilinear  velocity  v. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  I,  art.  281.) 


MECHANICS  11 

RADIUS   OF  GYRATION,    (i) 

The  radius  of  gyration  of  any  body  (made  up  of  a 
system  of  particles)  about  any  axis  is  the  perpendicular 
distance  from  that  axis  at  which,  if  the  whole  mass  were 
placed,  it  would  have  the  same  moment  of  inertia  as  before. 

Mathematically  the  radius  of  gyration  is, 


The  kinetic  energy  of  rotation  of  the  body  when  its 
angular  velocity  is  w  is: 

Er=  3^o>2Zmr2=  3/£«2k22m. 

(Consult  Thomson  and  Tait,  Treatise  on  Natural  Phi- 
losophy, Part  I,  art.  281.) 

*> 

RADIUS   OF  GYRATION,    (a) 

If  the  radius  of  gyration -of  any  body  revolving  about  a 
line  through  its  center  of  mass  is  k,  then  the  radius  of 
gyration  K  about  any  parallel  line  a  distance  a  from  this 
line  is  given  by  the  relation, 

K2  =  k2  +'  a2. 
(Consult  Jeans,  Theoretical  Mechanics,  p.  291.) 


IS  LAWS  OF  PHYSICAL  SCIENCE 

ANALOGUES  IN  TRANSLATION  AND  ROTATION. 

Force  is  measured  by  the  product,  mass  X  linear  acceler- 
ation, or  F  =  ma. 

Moment  of  force  is  measured  by  the  product,  moment  of 
inertia  X  angular  acceleration,  or  Fr  =  la. 


One-half  the  mass  X  linear  velocity  =  kinetic  energy  of 
translation,  or  Et  =  %  mv2. 

One-half  the  moment  of  inertia  X  the  angular  velocity 
=  kinetic  energy  of  rotation,  or  Er  =  %  I  w2. 

When  no  external  forces  are  acting,  the  product,  mass 
X  velocity  is  constant,  or  mv  =  linear  momentum  =  a 
constant. 

Under  similar  conditions  the  product,  moment  of  inertia 
X  angular  velocity  is  constant,  or  I  w  =  angular  momentum 
=  a  constant. 

(Ames,  Theory  of  Physics,  Chap.  II.) 

SIMPLE  HARMONIC  MOTION. 

A  body  is  said  to  oscillate  with  Simple  Harmonic  Motion 
when  its  acceleration  is  always  directed  toward  the  middle 
point  of  its  path  of  oscillation  and  is  proportional  to  its 
displacement  therefrom. 

The  acceleration  is  given  by  the  formula, 


where  N  is  the  frequency  or  number  of  complete  oscillations 
in  the  unit  of  time  and  S  is  the  displacement  of  the  point 
from  its  middle  position  at  any  instant. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  I,  art.  57,) 


MECHANICS  13 

SIMPLE  PENDULUM. 

The  oscillation  of  a  pendulum  may  be  regarded  as  Simple 
Harmonic  Motion  when  the  amplitude  of  the  oscillation  is 
small.  The  approximate  time  of  complete  vibration  is, 


Here  1  =  the  length  of  the  simple  pendulum  and  g  = 
the  acceleration  of  gravity. 

(Jeans,  Theoretical  Mechanics,  pp.  259-262.  Also 
Ganot's  Physics,  art.  56.) 

LAW  OF  THE  COMPOUND  PENDULUM. 

If  y  is  the  length  of  a  simple  pendulum  which  oscillates 
in  the  same  time  as  a  compound  pendulum,  then  the  prin- 
ciple of  the  center  of  oscillation  asserts  that 

Jmr2 
y~  2mr" 

The  compound  pendulum  executes  isochronous  oscilla- 
tions when  the  amplitude  of  the  oscillations  is  small.  The 
time  of  a  complete  oscillation  is  that  of  a  simple  pendulum 
having  this  length  y. 

Here  2  mr  is  the  sum  of  the  products  of  the  elements 
of  mass  of  the  pendulum  multiplied  by  their  distances  from 
the  point  of  support  and  2  mr2  is  the  moment  of  inertia  of 
the  pendulum  about  its  axis  of  oscillations.  Its  complete 
period  for  oscillations  of  small  amplitude  is 


T  =  27r-mi 

g2mr 
(Mach,  Science  of  Mechanics,  pp.  173-177.) 


4 

* 


14  LAWS  OF  PHYSICAL  SCIENCE 

CONVERTIBILITY    OF    CENTER    OF   OSCILLATION    AND 
POINT    OF   SUSPENSION. 

When  the  center  of  oscillation  and  the  point  of  suspen- 
sion of  a  compound  pendulum  are  interchanged  the  time  of 
oscillation  remains  the  same. 

This  principle  was  employed  by  Captain  Kater  for 
determining  the  exact  length  of  the  seconds  pendulum. 
Hence  the  term  " Kater 's  Pendulum." 

(Mach,  Science  of  Mechanics,  p.  172  et  seq.  and  p.  186. 
Also  Ganot's  Physics,  art.  81.) 

CYCLOIDAL   PENDULUM. 

A  particle  subjected  to  the  force  of  gravity  and  con- 
strained to  move  in  a  cycloidal  path  will  have  a  harmonic 
motion  which  is  strictly  isochronous  whatever  its  amplitude 
of  oscillation. 

The  period  will  be, 

2D 
g 

where  D  is  the  diameter  of  the  rolling  circle  which  generates 
the  cycloid.  Thus  the  period  is  that  of  a  simple  pendulum  of 
length  2D. 

(Jeans,  Theoretical  Mechanics,  p.  267.) 

MINIMUM  POTENTIAL   ENERGY. 

When  a  system  is  in  stable  equilibrium  its  potential 
energy  is  as  small  as  possible ;  that  is,  any  small  movement 
imparted  to  the  system,  subject  to  its  degrees  of  freedom, 
will  increase  and  not  decrease  its  potential  energy. 

(For  a  mathematical  discussion  see  Jeans,  Theoretical 
Mechanics,  p,  174  et  seq.) 


MECHANICS  15 

IMPACT  BETWEEN  TWO  BODIES. 

When  two  bodies  meet  in  impact  the  impulse  of  restitu- 
tion T  equals  the  impulse  of  compression  I  times  a  certain 
coefficient  e  called  the  coefficient  of  elasticity  or  coefficient  of 
resilience. 

Thus,  I'  =  el. 

The  coefficient  e  is  unity  for  perfectly  elastic  bodies  and 
zero  for  perfectly  inelastic  bodies. 

Its  value  for  actual  bodies  must  be  obtained  by 
experiment. 

(Jeans,  Theoretical  Mechanics,  pp.  238-241.) 

NEWTON'S  EXPERIMENTAL  LAW  OF  IMPACT. 

When  two  bodies  meet  in  impact  and  their  centers  of 
gravity  lie,  at  the  moment  of  impact,  in  a  line  through  the 
point  of  contact,  then  the  normal  component  of  relative 
velocity  of  their  centers  of  gravity  after  impact  is  equal  to 
the  relative  velocity  before  impact  times  the  coefficient  of 
resilience,  and  is  in  the  opposite  direction,  or 

v  -  v'  =  -  e  (u  -  u') 

where  v,  v'  are  the  velocities  of  the  two  bodies  after  impact, 
u,  u'  their  velocities  before  impact  and  e  the  coefficient  of 
resilience. 

(Jeans,  Theoretical  Mechanics,  pp.  244,  245.) 

RELATION    CONNECTING  VELOCITIES   OF  TWO  BODIES 
BEFORE  AND  AFTER  IMPACT. 

If  two  bodies  of  masses  m  and  m'  have  velocities  u,  u' 
before  impact  and  velocities  v,  V  after  impact,  then  by  the 
conservation  of  momentum 

mv  +  mV  =  mu  +  m'u', 
or,  if  U  is  the  common  velocity  of  the  two  bodies  after  impact, 

U=     mu  +  m'u/     (called  the  rule  of  Wallis). 

m  +  m' 

(Jeans,  Theoretical  Mechanics,  p.  245.  Also  Mach, 
Science  of  Mechanics,  p.  318.) 


16  LAWS  OF  PHYSICAL  SCIENCE 

RELATION  OF  VELOCITIES  AFTER  AND  BEFORE  IMPACT. 

The  normal  velocities  v,  v'  of  two  impacting  bodies  of 
masses  m,  m'  after  collision  are  given  in  terms  of  their 
velocities  u,  u'  before  collision  by  the  two  relations, 

mu  +  m'u'  —  em'  (u  —  u') 

m  +  m'  ,  ' 

,  _    mu  +  m'u'  +  em  (u  —  u') 
m  +  m' 

where  e  is  the  coefficient  of  resilience. 
(Jeans,  Theoretical  Mechanics,  p.  245.) 

IMPACT. 

Relative  velocity  of  approach  and  recession  of  impinging 
perfectly  elastic  bodies  is  the  same. 

Also,  mV  +  mv  =  m'u'  +  mu 
and,  mV2  +  mv2  =  m'u'2  +  mu2, 

namely,  the  quantity  of  motion  before  and  after  impact, 
estimated  in  the  same  direction  is  the  same,  also,  the  vis 
viva,  or  kinetic  energy  of  the  system. 

(Consult  Mach,  Science  of  Mechanics,  p.  322.  See  Thom- 
son and  T'ait,  Treatise  on  Natural  Philosophy,  Part  I,  arts. 
300-301,  for  comments  upon  conditions  which  determine 
loss  of  kinetic  energy  with  impact.) 

TIME    OF   ELECTRICAL    CONTACT    OF    IMPACTING    STEEL-SPHERES. 

When  two  equal  steel-spheres  come  together  the  time  in 
millionths  of  a  second  during  which  they  are  in  contact 
equals  74.7  times  the  diameter  (in  cm.)  divided  by  the  fifth 
root  of  their  velocity  of  approach. 

Thus,  T/*  =  74.7  _5- 

v1/5 

where  T^t  =  microseconds, 

D   =  diameter  of  spheres  in  centimeters, 

v    =  velocity   in   centimeters  per  second   with 


MECHANICS  17 

which  they  approach  each  other  at  the  moment  of  the 
beginning  of  contact. 

(This  law  was  experimentally  determined  by  E,  F. 
Northmp  and  A.  E.  Kennelly.  See  Jour.  Frank.  Inst., 
July,  1911.) 

VELOCITY  OF  A  DISTURBANCE  IN  AN  ELASTIC  MEDIUM. 

The  velocity  with  which  any  compressional  disturbance 
is  propagated  through  an  elastic  medium  is  given  by  the 
equation 


where  E  is  the  coefficient  of  volume  elasticity  and  D  is  the 
density  of  the  medium.    This  law  is  due  to  Newton. 
(Watson,  A  Text  Book  of  Phyvics,  pp.  362-365.) 

BERTRAND'S  PRINCIPLE    OF   SIMILITUDE. 

If  two  systems  differ  only  in  geometrical  magnitude, 
the  masses  of  corresponding  parts  being  proportional  to 
each  other,  they  will  be  mechanically  similar  if  the  forces 
acting  on  and  the  velocities  of  respective  parts  of  the  systems 
bear  the  relation 

2 

force  proportional  to  mass  X  velocity    . 
linear  dimensions 

This  principle  gives  the  conditions  under  which  two 
systems  which  are  geometrically  similar  may  also  be  mechan- 
ically similar  and  is  important  in  testing  small  model 
machines. 

(Routh,  Elementary  Rigid  Dynamics,  pp.  234-297.  For 
a  full  discussion,  see  article  by  Bertrand  in  Cahier  32,  of  the 
Journal  de  I' E  cole  Poly  technique.) 


18  LAWS  OF  PHYSICAL  SCIENCE 

CHANGE  OF  MOMENTUM  DUE  TO  AN  IMPULSIVE  FORCE. 

When  a  body  moves  in  such  a  way  that  its  configuration 
with  respect  to  the  force  which  acts  upon  it  remains  always 
the  same,  the  moving  force  is  measured  by  the  rate  of 
increase  of  the  momentum. 

If  F  is  the  force,  p  the  momentum  and  t  the  time, 

F  =     ,f       whence 
at 

p  = jFdt. 

This  time-integral  is  called  the  Impulse  of  the  Force. 
(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  558.) 

SLIDING  FRICTION. 

The  tangential  force  required  to  slide  one  body  over 
another  is  independent  of  the  extent  of  their  surfaces  in 
contact  and  of  the  rate  of  sliding.  It  is  proportional  to  the 
normal  force  R  pressing  them  together  and  to  a  constant 
for  the  two  bodies  called  the  coefficient  of  friction  p,.  Thus 
F  =  p  R. 

Distinction  is  made  between  statical  friction  where  a 
relative  motion  of  the  bodies  is  just  not  produced  by  the 
acting  forces  and  kinetic  friction  where  there  is  relative 
motion  of  the  bodies  in  contact.  Kinetic  friction  is  generally 
less  than  the  extreme  force  of  static  friction. 

(Thomson  and  Tait,  Treatise  on  Natural  Philosophy, 
Part  II,  arts.  450-452.) 

ROLLING    FRICTION. 

Rolling  friction  acts  much  as  does  sliding  friction.  The 
coefficient  u  is  in  this  case  much  smaller  and  the  force 
required  to  cause  motion  is  again  proportional  to  the  normal 
force  R  pressing  the  bodies  together. 

Thus  F  =  uR. 

(Ganot's  Physics,  art.  49.) 


MECHANICS  19 

HOOKE'S  LAW. 

Hooke's  law  states  that  within  the  elastic  limit  of  any 
body  the  ratio  of  the  stress  acting  upon  the  body  to  the 
strain  produced  is  constant.  This  constant  ratio  of  stress 
to  strain  for  any  particular  type  of  change  in  any  body  is 
called  its  " coefficient  of  elasticity." 

(Ames,  Theory  of  Physics,  pp.  103-104.  Also  see 
Ganot's  Physics,  art.  85.) 

YOUNG'S  MODULUS. 

By  "Young's  Modulus"  is  understood  the  force  Which 
would  be  required  to  stretch  a  body  of  unit  cross-section  to 
double  its  length  if  such  lay  within  the  elastic  limit.  It 
is  a  constant  for  any  one  material. 

T  P1 

Thus,  Young's  Modulus  =  M  =• 


ea 

where  L  =  length  of  body  .(usually  a  wire), 
a  =  area  of  cross-section, 
F==  stretching  force  applied  and 
e  =  total  elongation  produced  by  F. 
Similar  moduli  hold  for  compression,  flexure  and  torsion. 
(Crew,  General  Physics,  p.  131.     Also  Ganot's  Physics, 
art.   87.     Also   Thomson   and   Tait,   Treatise  on  Natural 
Philosophy,  Part  II,  arts.  686-688.) 

CONSERVATIVE    SYSTEMS. 

A  system  is  called  a  conservative  system  when  it  is  such 
that  the  total  work  done  in  performing  any  series  of  dis- 
placements which  bring  the  system  back  to  its  original  con- 
figuration is  algebraically  zero. 

(Consult  Thomson  and  Tait,  Treatise  on  Natural  Phi- 
losophy, Part  I,  art,  271  et  seq.) 


20  LAWS  OF  PHYSICAL  SCIENCE 

PRINCIPLE  OF  WORK  DONE  BY  A  CONSERVATIVE  SYSTEM. 

When  a  conservative  system  is  changed  from  one  con- 
figuration to  another  the  work  done  is  independent  of  the 
manner  in  which  the  change  is  made  and  depends  only  on  the 
initial  and  final  states  of  the  system. 

(Jeans,  Theoretical  Mechanics,  p.  164.) 

RAYLEIGH'S  RECIPROCATION  THEOREM. 

If  a  system  of  bodies  is  struck  successively  at  two  dif- 
ferent points  a  and  b  by  impulsive  forces  Px  and  Q2,  each 
blow  will  in  general  affect  all  the  bodies  of  the  system.  Let 
u±,  u2  be  the  velocities  of  the  points  a  and  b  produced  by 
the  blow  P±  and  v1?  v2  be  the  velocities  produced  by  the 
blow  Q2.  Eayleigh's  Theorem  states  that 
PI  v±  =  Q2  u2. 

(Routh,  Elementary  Rigid  Dynamics,  pp.  338,  339. 
Many  illustrations  are  given  in  Eayleigh's  Theory  of 
Sound.) 

PERPETUAL  MOTION  IMPOSSIBLE. 

The  impossibility  of  perpetual  motion  rests  upon  the 
following  consideration :  Let  a  system  be  made  to  pass  by 
frictionless  constraint  from  a  configuration  A  to  another 
configuration  B  and  return  by  another  path  to  A.  Assume 
more  work  is  done  upon  the  system  by  the  mutual  forces  when 
it  follows  one  path  than  when  it  follows  the  other.  Let 
this  process  be  repeated  over  and  over  again  forever.  The 
system  will  then  be  a  continual  source  of  energy  without  the 
consumption  of  materials,  something  which  all  experience 
shows  to  be  impossible. 

(Consult  Thomson  and  Tait,  Treatise  on  Natural  Phi- 
losophy, Part  I,  art.  272.) 

CONSERVATION  OF  THE  MOVEMENT  OF  THE  CENTER  OF  GRAVITY. 

The  state  of  rest  or  movement  of  the  center  of  gravity 
of  several  bodies  is  not  altered  by  the  reciprocal  action  of 


MECHANICS  21 

these  bodies.  Or,  the  motion  of  the  center  of  gravity  of  a 
system  of  bodies  is  changed  only  by  forces  external  to  the 
system. 

(For  extension  of  principle  consult  Laplace,  Mecanique 
Celeste,  Vol.  I,  and  Lagrange,  Mecanique  Analytique,  Vol. 
I,  Part  II,  Section  1,  art.  15,  and  Section  3,  art.  1  et  seq.) 

PRINCIPLE  OF  THE  MOTION  OF  THE  CENTER  OF  GRAVITY. 

The  center  of  gravity  of  any  system  moves  as  if  all 
the  external  forces  were  acting  on  the  entire  mass  of  the 
system  concentrated  in  a  particle  at  the  center  of  gravity 
and  all  the  external  forces  were  applied  to  this  particle. 

(Jeans,  Theoretical  Mechanics,  pp.  224-226.) 

CONSERVATION  OF  MATTER  OR  MASS. 

If  a  mass  or  quantity  of  matter  of  any  kind  be  selected 
for  consideration,  then  the  ' '  principle  of  the  conservation  of 
mass"  asserts;  that  no  changes  of  any  kind  which  can  occur 
in  the  mass,  whether  brought  about  by  the  action  of  forces 
internal  or  external  to  the  mass,  can  alter  in  the  slightest 
degree  its  total  quantity. 

For  the  measure  of  the  total  quantity  of  mass  considered, 
may  be  taken,  either  its  inertia  or  the  gravitational  attrac- 
tion of  the  earth  upon  it  when  located  at  a  particular  place. 

(Consult  Ames,  Theory  of  Physics,  pp.  6,  7.  Also 
see  definition  under  word  CONSERVATION,  New  Century 
Dictionary.) 

LAW   OF   CONSERVATION    OF   MOMENTUM.     (l) 

When  any  system  of  particles  moves  without  being  acted 
upon  by  any  external  force,  the  total  momentum  of  the 
system  remains  constant  in  magnitude  and  direction,  and 
the  moment  of  momentum  about  any  axis  remains  constant. 

(Consult  Mach,  Science  of  Mechanics,  p.  288.) 


«  LAWS  OF  PHYSICAL  SCIENCE 

LAW    OF   CONSERVATION    OF   MOMENTUM,     (a) 

If  all  velocities  in  a  given  direction  are  reckoned  as 
positive  and  all  in  the  opposite  direction  as  negative,  then 
the  sum  of  the  momenta  of  a  system  of  bodies,  uninfluenced 
by  forces  external  to  the  system,  is  preserved  constant, 
whether  or  not  these  bodies  meet  in  impact  and  whether 
or  not  they  are  elastic  or  inelastic, 

(  For  mathematical  definition,  ^m  — J  =  a  constant,  see 

dt 

Appell,  Traite  de  Mecanique  Rationnelle,  Vol.  II,  p.  20. 
Consult  Mach,  Science  of  Mechanics,  p.  326.) 

LAW  OF  THE  CONSERVATION  OF  MOMENT  OF  MOMENTUM. 

Let  a  system  of  particles  of  masses  m^,  m2,  m3,  etc.,  have 
velocities  ul9  u2,  u3,  etc.,  in  a  plane  or  in  planes  which  are 
parallel.  If  O  be  the  position  of  any  fixed  line  perpen- 
dicular to  these  planes  and  rly  r2,  rs,  etc.,  the  perpendicular 
distances  from  this  line  to  the  directions  in  which  the  various 
particles  are  moving  at  a  given  instant  the  products,  m  u  r, 
are  "moments  of  momenta"  and  the  law  of  the  conservation 
of  moment  of  momentum  states  that  the  I'mur  =  a  con- 
stant, if  no  forces  external  to  the  system  act  upon  it.  In 
taking  the  sum  of  the  products,  the  products  are  reckoned 
positive  or  negative  according  as  a  particle  viewed  from  0 
is  moving  toward  the  right  or  toward  the  left. 

(Ames,  Theory  of  Physics,  p.  49.  Consult  also  La- 
grange,  Mecanique  Analytique,  Vol.  I.,  Part  II,  Section  1, 
art.  16,  p.  260.) 

CONSERVATION   OF  LIVING  FORCES. 

This  principle  states  that  the  difference  of  the  force- 
functions  (or  work)  at  the  beginning  and  at  the  end  of  the 
motion  of  a  system  is  equal  to  the  difference  of  the  vires 


MECHANICS  23 

vivce   (kinetic  energies)   at  the  beginning  and  the  end  of 
the  motion.    Namely, 

j(U-U0)  =  ^Km(v2-v§). 

(For  a  formal  development  of  this  principle  see  Mach, 
Science  of  Mechanics,  p.  478  et  seq.) 

PRINCIPLE  OF  THE  CONSERVATION  OF  AREAS 
(D'ARCY'S  STATEMENT). 

According  to  d'Arcy,  the  principle  states  that  the 
sum  of  the  products  of  the  mass  of  each  body  by  the  area 
which  its  radius  vector  describes  about  a  fixed  center  on 
any  plane  of  projection  is  always  proportional  to  the 
time.  Or, 

SmA  =  Kt       or  Jm  xdy"ydx  =K  dt. 

(See  Laplace,  Mecanique  Celeste,  Vol,  I,  Part  I,  Book  I, 

art.  21.) 

PRINCIPLE   OF  THE   CONSERVATION  OF  AREAS 
(MACK'S   STATEMENT). 

Mach  states  this  principle  as  follows:  If  from  any 
point  in  space  radii  be  drawn  to  several  masses  and  projec- 
tions be  made  upon  any  plane  of  the  areas  which  the 
several  radii  describe,  the  sum  of  the  products  of  these  areas 
into  the  respective  masses  will  fre  independent  of  the  action 
of  internal  forces. 

(Mach,  Science  of  Mechanics,  p.  294.) 


24  LAWS  OF  PHYSICAL  SCIENCE 

PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

If  the  point  of  application  of  a  force  be  displaced  through 
a  small  space,  the  resolved  part  of  the  displacement  in  the 
direction  of  the  force  has  been  called  its  "Virtual  Velocity.  " 
Mathematically  expressed  the  principle  asserts  that  for  a 
body  to  be  in  equilibrium 

Pp  +  P'p'  +  P"p"-H  ......  =  O. 

Here  P,  P',  P",  etc.,  are  applied  forces  acting  upon  a 
connected  system  at  points  A,  B,  C,  etc.,  and  p,  p',  p",  etc., 
are  the  projections  on  the  lines  of  the  forces  of  small  dis- 
placements of  the  points.  The  projections  are  to  be  taken 
positive  when  they  fall  in  the  direction  of  a  force  and 
negative  when  they  fall  in  the  opposite  direction. 

(See  Mach,  Science  of  Mechanics,  Chap.  I,  Sec.  IV,  p. 
49  et  seq.  for  a  clear  and  full  discussion  of  this  principle.) 

PRINCIPLE  OF  LEAST  ACTION. 

In  the  movement  of  bodies  which  interact  so  that  the 
total  energy  remains  constant,  the  sum  of  the  products  of 
the  masses  by  the  velocities  and  by  the  spaces  described 
is  a  minimum.  This  principle  was  extended  to  systems  of 
masses  by  Lagrange  who  presented  it  in  the  form, 


(See  Mach,  Science  of  Mechanics,  pp.  364-380  for  a  good 
physical  discussion  of  this  rather  obscure  principle.  It  is 
mathematically  treated  in  Thomson  and  Tait,  Treatise  on 
Natural  Philosophy,  Part  1,  arts.  326,  327.) 

KELVIN'S    MINIMUM-ENERGY    THEOREM. 

If  a  material  system,  initially  at  rest,  is  set  in  motion  by 
impulses  applied  to  particular  points  in  such  a  way  that 
these  points  acquire  specified  velocities,  the  motion  of  the 
entire  system  is  such  as  to  make  the  total  kinetic  energy  of 


MECHANICS  25 

the  system  less  than  it  would  be  in  any  other  possible  motion 
of  the  system  consistent  with  the  same  velocity  conditions. 
(Houstoun,  An  Introduction  to  Mathematical  Physics, 
p.  67.) 

STOKES'  LAW  FOR  THE  FALL  OF  A  SPHERE  THROUGH  A 
VISCOUS    MEDIUM. 

When  a  small  sphere  falls  under  the  action  of  gravity 
through  a  viscous  medium  it  ultimately  acquires  a  constant 
velocity  equal  to 

_  Xga'(di-d») 

k 

where  a  is  the  radius  and  dt  the  density  of  the  sphere,  d, 
is  the  density  of  the  medium  and  k  is  its  coefficient  of 
viscosity. 

When  great  accuracy  is  required,  correction  factors  must 
be  added  to  the  above  expression. 

This  formula  has  been  of  much  service  in  determining 
the  charge  on  an  electron. 

(G-.  GL  Stokes,  Mathematical  and  Physical  Papers,  Vol. 
Ill,  p.  59.  See  also  Campbell,  Modern  Electrical  Theory, 
P.  91.) 

D'ALEMBERT'S  PRINCIPLE,     (i) 

When  forces  act  upon  one  or  more  rigidly  connected 
points  of  a  system  of  masses,  these  forces,  called  the  im- 
pressed forces,  may  each  be  resolved  into  two  components, 
the  equilibrated  forces  and  the  effective  forces.  The  latter 
only  are  operative  in  producing  motion,  while  the  former 
form  a  system  balanced  by  the  connections.  The  sum  of 
the  products  of  the  effective  forces  by  the  elementary  dis- 
placements which  they  produce  is  equal  to  the  element  of 
work  performed  upon  the  system. 

(Mach,  Science  of  Mechanics,  pp.  335-343.) 


26  LAWS  OF  PHYSICAL  SCIENCE 

D'ALEMBERT'S  PRINCIPLE,     (a) 

All  of  the  work  performed  on  any  system  is  performed 
by  the  effective  components  of  the  impressed  forces.  When 
no  work  is  performed  the  system  is  in  equilibrium.  The 
two  mathematical  forms  in  which  D'Alembert's  principle 
is  usually  expressed  are: 

J[(X-ma)  Sx  +  (Y-mb)  5y  +  (Z-mc)  6z]  =  O, 
2  (XSx  +  Y5y  +  Z5z)  =  ^m  (a5x  +  b5y  +  cSz). 

Here  X,  Y,  Z  are  the  mutually  perpendicular  com- 
ponents parallel  to  rectangular  coordinates  of  every  force  P 
impressed  on  the  masses  m.  ma,  mb,  me  are  the  corre- 
sponding components  of  every  effective  force  W,  where  a, 
b,  c  denote  accelerations  and  8x,  8y,  8z  are  displacements  in 
the  directions  of  the  coordinates. 

(Consult  Mach,  Science  of  Mechanics,  p.  342.) 

GAUSS'S  PRINCIPLE   OF  LEAST   CONSTRAINT,     (i) 

Let  the  masses  M,  M^  etc.,  be  joined  in  any  manner  with 
one  another.  If  these  masses  were  free  they  would  describe 
in  the  element  of  time  under  the  action  of  forces  impressed 
on  them,  paths  ab,  a^,  etc.;  but  in  consequence  of  the 
connections,  they  describe  in  the  same  element  of  time  the 
paths  ac,  a^,  etc.  Gauss's  principle  asserts  that  the  motion 
of  the  connected  points  is  such  that,  for  the  motion  actually 
taken,  the  sum  of  the  products  of  the  mass  of  each  particle 
into  the  square  of  the  distance  of  its  deviation  from  the 
position  it  would  have  reached  if  free,  is  a  minimum. 

(Consult  Mach,  Science  of  Mechanics,  p.  350  et  seq.  for 
a  clear  development  of  this  principle.) 

GAUSS'S  PRINCIPLE   OF  LEAST   CONSTRAINT,     (a) 

The  motion  of  a  system  of  material  points  intercon- 
nected in  any  way  and  submitted  to  any  influences,  accords 
at  each  instant  as  closely  as  possible  with  the  motion  the 


MECHANICS  27 

points  would  have  if  they  were  free.  The  actual  motion 
takes  place  so  that  the  constraints  on  the  system  are  the 
least  possible.  For  the  measurement  of  the  constraint, 
during  any  element  of  time,  is  to  be  taken  the  sum  of  the 
products  of  the  mass  of  each  point  by  the  square  of  its 
deviation  from  the  position  it  would  have  occupied  at  the 
end  of  the  element  of  time,  if  it  had  been  free. 

PRINCIPLE   OF  LEAST    CONSTRAINT;  COMMENT  ON. 

Gauss's  principle  of  "least  constraint"  gives  equations 
which  when  differentiated  yield  D'Alembert's  principle. 
The  kernel  idea  is  "that  the  work  of  the  forces  which 
deviate  the  movement  of  the  system  from  the  paths  it  would 
take  if  unconstrained  is  as  small  as  possible  under  the  con- 
ditions." Gauss's  principle  includes  both  statical  and 
dynamical  cases. 

(Clear  expositions  and  analytical  treatments  of  the 
principle  of  least  constraint  are  to  be  found  in  Appel, 
Traite  de  Mecanique  Rationnelle,  and  in  Mach,  Science  of 
Mechanics. ) 

HAMILTON'S  PRINCIPLE. 

This  principle  is  expressed  as  follows: 

dt  =  °'  or  Jt!1  (5U  +  5T)  dt  =  °' 
where  SU  and  8T  denote  the  variations  of  the  work  and  vis 
viva  vanishing  for  the  initial  and  terminal  epochs.  '  *  Hamil- 
ton's  principle  is  easily  deduced  from  D ' Alembert 's,  and, 
conversely,  D'Alembert's  from  Hamilton's."  The  prinj- 
ciples,  least  action,  least  constraint,  D'Alembert's,  Gauss's 
and  Hamilton's,  are  not  expressions  of  different  facts,  but 
rather,  are  simply  views  of  different  aspects  of  the  same 
fact. 

(Mach,  Science  of  Mechanics,  pp.  380-384.  Also  consult 
Appell,  Traite  de  Mecanique  Rationnelle,  Vol.  II,  p.  422.) 


28  LAWS  OF  PHYSICAL  SCIENCE 

PRINCIPLE   OF   THE   CONSERVATION   OF   ENERGY. 

In  every  modification  of  a  material  system,  not  affected 
by  forces  foreign  to  the  system,  the  sum  of  its  potential 
and  kinetic  energies  remains  constant. 

Calling  E  the  kinetic  energy  and  P  the  potential  energy 
of  the  system,  E  +  P  =  K,  a  constant. 

(See  Helmholtz's  lecture  delivered  at  Carlsruhe  about 
1862,  "On  the  Conservation  of  Force,"  in  Popular  Scien- 
tific Lectures,  Vol.  I.  Also  consult  Thomson  and  Tait, 
Treatise  on  Natural  Philosophy,  Part  I,  arts.  269-278.) 

DIFFERENT   STATES   OF   EQUILIBRIUM. 

A  body  is  in  stable  equilibrium  when  a  slight  movement 
from  its  position  will  raise  its  center  of  gravity.  It  is  in 
unstable  equilibrium  when  such  movement  will  lower  its 
center  of  gravity.  It  is  in  neutral  equilibrium  when  such 
movement  will  neither  raise  nor  lower  its  center  of  gravity. 

(Ganot's  Physics,  art.  72.) 


II 

HYDROSTATICS,  HYDRODYNAMICS  AND 
CAPILLARITY 


HYDROSTATICS,  HYDRODYNAMICS  AND 
CAPILLARITY 

ARCHIMEDES'   PRINCIPLE. 

When  a  body  is  in  equilibrium  in  a  fluid  the  fluid  exerts 
an  upward  force  on  the  body  equal  to  the  weight  of  the 
displaced  fluid  and  acts  through  its  center  of  gravity. 
Or  a  body  immersed  in  liquid  loses  weight  equal  to  the 
weight  of  the  displaced  liquid. 

(This  principle  furnishes  the  most  convenient  method 
for  the  determination  of  the  specific  gravity  of  a  body,  in 
terms  of  that  of  the  fluid  used.) 

(Ganot's  Physics,  art.  112.) 

EQUILIBRIUM  OF  FLOATING  BODIES. 

1.  The  floating  body  must  displace  a  volume  of  liquid  whose 

weight  equals  that  of  the  body. 

2.  The  center  of  gravity  of  the  floating  body  must  be  in  the 

same  vertical  line  with  that  of  the  fluid  displaced. 

3.  The  equilibrium  of  a  floating  body  is  stable  or  unstable 

according  as  the  metacenter  is  above  or  below  the 
center  of  gravity. 
(Ganot's  Physics,  art.  114.) 

PRESSURE  PRODUCED  IN  LIQUIDS  BY  GRAVITY. 

The  pressure  in  each  layer  is  proportional  to  the  depth. 

With  different  liquids  and  the  same  depth,  the  pres- 
sure is  proportional  to  the  density  of  the  liquid. 

The  pressure  is  the  same  at  all  points  of  the  same 
horizontal  layer. 

(Ganot's  Physics,  art.  98.) 

31 


32  LAWS  OF  PHYSICAL  SCIENCE 

HYDROSTATIC    PARADOX. 

The  total  weight  or  downward  force  exerted  by  a  vessel 
containing  liquid  depends  on  the  shape  and  size  of  the 
containing  vessel  and  may  be  greater  or  smaller  than  the 
force  which  is  applied  to  give  the  liquid  its  hydrostatic 
pressure. 

(Consult  Ganot's  Physics,  art.  102.  Also  Ames,  Theory 
of  Physics,  pp.  112-114.) 

PASCAL'S  LAW. 

The  fluid  pressure  due  to  the  reaction  of  the  walls  of 
the  containing  vessel  is  the  same  at  all  points  throughout 
the  fluid.  Or  pressure  exerted  anywhere  upon  a  mass  of 
liquid  is  transmitted  undiminished  in  all  directions,  and 
acts  with  the  same  force  on  all  equal  surfaces  and  in  a 
direction  at  right  angles  to  those  surfaces. 

(Kimball,  College  Physics,  p.  111.  Also  Ganot's  Physics, 
art,  97.) 

CONDITION   OF   THE   EQUILIBRIUM    OF   LIQUIDS. 

1.  Its  surface  must  be  everywhere  perpendicular  to  the 

resultant  of  the  forces  which  act  on  the  molecules  of 
the  liquid. 

2.  Every  molecule  of  the  mass  of  the  liquid  must  be  subject 

in  every  direction  to  equal  and  contrary  forces. 
(Consult    Ganot's   Physics,   art.  103.     For  Clairaut's 
mathematical  statement  of  the  general  condition  of  liquid 
equilibrium,  see  Mach,  Science  of  Mechanics,  p.  397.) 

EQUILIBRIUM  OF  LIQUIDS  IN  COMMUNICATING  VESSELS. 

When  two  non-miscible  liquids  of  different  densities  are 
placed  in  communicating  vessels  their  free  surfaces  will 
stand  at  different  heights  above  the  surface  of  contact  of  the 
two  liquids.  Neglecting  any  capillary  action,  the  heights 


HYDROSTATICS,  HYDRODYNAMICS  AND  CAPILLARITY    33 

of  the  free  surfaces  of  the  two  liquids  above  Iheir  surface 
of  contact  are  in  inverse  ratio  to  their  densities. 

(Ames,  Theory  of  Physics,  p.  115.  Also  Chwolson,  Traite 
de  Physique,  Vol.  I,  Part  5,  p.  569.)- 

RESULTANT  OF  FORCES  OF  COHESION  AT  THE  SURFACE  OF  A 
LIQUID. 

At  the  surface  of  a  liquid  all  the  forces  of  cohesion  have 
a  resultant  which  is  directed  toward  the  interior  of  the 
liquid  normally  to  its  surface. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  p.  563.) 

EFFLUX:  TORRICELLI'S   THEOREM. 

The  velocity  of  efflux  of.  a  stream  of  liquid  issuing  from 
a  cistern  is  the  velocity  which  a  freely  falling  body  would 
have  on  reaching  the  orifice  after  having  started  to  fall 
from  rest  at  the  surface-level  of  the  fluid.  (Only  strictly 
true  when  friction  and  shape  of  the  orifice  are  disregarded. ) 

The  velocity  is, 

v  =  *s/2gh 
where  g  =  acceleration  of  gravity,  and 

h  =  height  of  level  of  liquid  above  orifice. 
(Ganot's  Physics,   art.   142.    Also  Mach,    Science   of 
Mechanics,  p.  402.) 

QUANTITY   OF   EFFLUX:   "VENA   CONTRACTA." 

The  stream  of  water  issuing  from  a  cistern  through  a 
circular,  sharp-edged  orifice  of  area  A  contracts  after  leav- 
ing the  orifice.  The  theoretical  discharge  would  be 
K  =  A^/^gh,  but  the  actual  discharge  is  found  to  be  about 
0.62  A-v/2gh,  where  g  equals  acceleration  of  gravity  and  h 
equals  height  of  level  of  liquid  above  orifice. 

The  reduced  velocity  is  due  to  the  contraction  of  the 
cross-section  of  the  stream,  called  the  vena  contracta. 

(Consult  Ganot's  Physics,  art.  145.  Also  Chwolson, 
Traite  de  Physique,  Vol.  I,  Part  5,  pp.  697,  698.) 


M  LAWS  OF  PHYSICAL  SCIENCE 

BERNOULLI'S  THEOREM:  FLOW  OF  LIQUIDS. 

At  any  point  in  a  tube,  through  which  a  liquid  is  flowing, 
the  pressure  plus  the  potential  energy  due  to  position  plus 
the  kinetic  energy  remains  constant  (friction  being  disre- 
garded). Or,  along  a  horizontal  stream-line,  the  relation 
holds, 

%  density  X  velocity  2+  pressure  =  a  constant. 
The  general  mathematical  statement  of  the  theorem  is, 

p  +  gph  +  %  pv2  =  c,  a  constant, 
where  p  =  density  of  liquid, 

g  =  acceleration  of  gravity, 
v  =  velocity  of  flow, 

h  =  distance  above  any  horizontal  plane  of  reference 
to  the  point  in  the  liquid  considered  and 

p  =  the  hydrostatic  pressure. 

(Consult  Crew,  General  Physics,  p.  149.  Also  Mach, 
Science  of  Mechanics,  p.  413  et  seq.  Also  Chwolson,  Traite 
de  Physique,  Vol.  I,  Part  5,  p.  690.) 

FLOW  OF  WATER  IW  PIPES. 

1.  The  loss  of  head  from  friction  is  proportional  to  the 

length  of  pipe. 

2.  It  increases  with  the  roughness  of  the  interior  surface. 

3.  It  decreases  as  the  diameter  of  the  pipe  increases. 

4.  It  increases  nearly  as  the  square  of  the  velocity. 

5.  It  is  independent  of  the  pressure  of  the  water. 

(Merriman,  Treatise  on  Hydraulics,  p.  209.) 


HYDROSTATICS.  HYDRODYNAMICS  AND  CAPILLARITY    85 
PRINCIPLE    OF   CONTINUITY. 

In  a  state  of  steady  flow,  the  quantity  of  fluid  passing 
any  cross-section  of  the  stream  in  a  given  time  is  the  same 
for  all  sections  of  the  stream.  Thus  the  product  SV  is 
constant,  where  S  is  the  cross-section  and  V  the  velocity 
of  the  stream  at  any  point. 

The  " Equation  of  Continuity/'  which  expresses  this 
fact  mathematically  is, 

JL  ^*  i    dlL  ,  dv   ,   dw      o 

5  dt  "*"  dx  "*"  dy  +  dz  " 

where  u,  v,  w  are  the  velocity-components  in  the  directions 
of  the  x,  y,  z  axes  respectively  at  any  point  of  the  fluid, 
8  is  the  density  of  the  fluid  and  t  represents  time. 

If  the  fluid  is  incompressible  the  first  term  of  this 
equation  vanishes. 

(Webster,  The  Dynamics  of  Particles  and  of  Rigid, 
Elastic  and  Fluid  Bodies,  pp.  496-499.) 

RESISTANCE  TO  THE  MOTION  OF  A  SOLID  THROUGH  A  FLUID. 

The  pressure  against  a  solid  moving  through  a  fluid  is 
approximately  proportional  to  the  square  of  their  relative 
velocity. 

(Values  of  the  constant  of  proportionality  for  the 
important  case  of  wind  against  a  sail  or  aeroplane  are  given 
in  Smithsonian  Physical  Tables,  p.  124.) 


36  LAWS  OF  PHYSICAL  SCIENCE 

VORTEX    MOTION. 

Definitions:  A  Vortex  Line  is  a  curve  whose  tangent  at 
every  point  coincides  with  the  direction  of  the  instantaneous 
axis  of  rotation  at  that  point.  A  space  bounded  by  vortex 
lines  is  called  a  Vortex  Tube  and  the  enclosed  fluid  is  said 
to  have  a  Vortex  Motion.  The  strength  of  such  a  tube  at 
any  cross-section  normal  to  its  axis  is  defined  as  the  product 
of  the  angular  velocity  w  and  the  cross-sectional  area  S  of 
the  tube  at  that  point. 

Laws: 

1.  A  vortex  tube  always  contains  the  same  elements  of 

fluid. 

2.  The  strength  wS  of  a  vortex  tube  is  the  same  at  all  parts 

of  the  tube  and  does  not  change  with  time    (in  a 
perfect  fluid.) 

3.  Vortex -tubes  are  either  closed  surfaces  or  have  their 

extremities  in  the  surface  of  the  fluid. 
(Webster,  The  Dynamics  of  Particles  and  of  Rigid, 
Elastic  and  Fluid  Bodies,  pp.  509-511.  For  a  complete 
treatment  see  Helmholtz,  Uber  Integrale  der  hydrodynam- 
ischen  Gleichungen,  welche  den  Wirbelbewegungen  ent- 
sprechen,  Wissenschaftliche  Abhandlungen,  Vol.  I,  p.  101. 
For  a  photographic  study  of  vortex  motions  in  water  see 
"An  Experimental  Study  of  Vortex  Motions  in  Liquids," 
by  E.  F.  Northrup,  Jour,  of  the  FrankUn  Institute,  Sept. 
and  Oct.,  1911.) 


HYDROSTATICS,  HYDRODYNAMICS  AND  CAPILLARITY     37 
FLOW  THROUGH  CAPILLARY  TUBES.    POISEUILLE'S  LAW. 

The  volume  of  liquid  V  which  will  flow  in  unit  time 
through  a  capillary  tube  of  length  1  and  radius  r,  is  given 

by  the  formula     v  =  ?^->      In  C.G.S.  units,  p  =  pressure- 
8  k  1 

difference  over  the  length  1  of  the  tube  in  dynes  per 
square  centimeter  and  k  is  the  coefficient  of  internal  fric- 
tion or  viscosity. 

The  reciprocal  of  the  viscosity,  namely -r-,  is  called  the 
fluidity. 

This  law  was  discovered  and  investigated  by  M.  Pois- 
euille  in  1843. 

(Consult  Poynting  and  Thomson,  Properties  of  Matter, 
pp.  207-209.  Also  Chwolson,  Traite  de  Physique,  Vol.  I, 
Part  5,  p.  674.) 

HYDRODYNAMICAL    THEOREM. 

"If  the  bounding  surface  of  a  liquid,  primitively  at 
rest,  be  made  to  vary  in  a  given  arbitrary  manner,  the  vis 
viva  of  the  entire  liquid  at  each  instant  will  be  less  than 
it  would  be  if  the  /liquid  had  any  other  motion  consistent 
with  the  given  motion  of  the  bounding  surface." 

(Mathematical  and  Physical  Papers,  by  Lord  Kelvin, 
Vol.  I,  p.  109.) 

CAPILLARY   ACTION:   TURIN'S    LAW. 

For  the  same  liquid  and  the  same  temperature,  the  mean 
height  of  the  ascent  in  a  capillary  glass-tube  is  inversely  as 
the  diameter  of  the  tube.  Thus  diameter  X  height  =  a 
constant. 

(Ganot's  Physics,  art.  131.  Also  consult  Poynting  and 
Thomson,  Properties  of  Matter,  p.  140.  Also,  Chwolson, 
Traite  de  Physique,  Vol.  I,  Part  5,  p.  616.) 


88  LAWS  OF  PHYSICAL  SCIENCE 

LAW  OF  CAPILLARY  ACTION,     (i) 

For  various  liquids  and  the  same  temperature  and  in 
tubes  of  the  same  diameter  the  mean  heights  to  which  the 
liquids  will  rise  vary  with  the  nature  of  the  liquid.  ( Of  all 
liquids  water  rises  the  highest.) 

(Ganot's  Physics,  art.  131.) 

LAW  OF  CAPILLARY  ACTION.     (2) 

For  the  same  liquid  and  the  same  temperature,  the  mean 
heights  to  which  the  liquid  rises  are  independent  of  the  form 
of  the  capillary  tube  except  at  the  meniscus.  Provided  the 
liquid  moistens  the  tube,  neither  the  thickness  of  the  tube 
nor  its  nature  has  any  influence  on  the  height  to  which  the 
liquid  rises. 

(Ganot's  Physics,  art.  131.) 

LAW  OF  CAPILLARY  ACTION.     (3) 

The  height  to  which  a  liquid  rises  in  a  capillary  tube 
diminishes  as  the  temperature  increases.  As  the  height 
becomes  less  the  meniscus  becomes  flattened.  When  the 
sides  of  the  tubes  are  not  moistened  Jurin's  law  holds 
approximately  for  the  depression  of  the  liquid. 

(Ganot's  Physics,  art.  131.) 

CAPILLARY  CORRECTIONS   OF  MERCURY   COLUMNS. 

*  *  The  height  of  the  meniscus  and  the  value  of  the  capil- 
lary depression  depend  on  the  bore  of  the  tubing,  on  the 
cleanliness  of  the  mercury  and  on  the  state  of  the  walls  of 
the  tube.  The  correction  is  negligible  for  tubes  with  diam- 
eters greater  than  about  2.5  cms." 

(See  Kaye  and  Laby,  Physical  and  Chemical  Constants, 
p.  17.  Or  see  Smithsonian  Physical  Tables,  p.  123.) 


HYDROSTATICS,  HYDRODYNAMICS  AND  CAPILLARITY    39 
SURFACE  TENSION  AND  WORK  OF  THE  FORCES  OF  COHESION. 

Every  diminution  in  the  extension  of  a  liquid-surface  is 
associated  with  work  done  by  forces  of  molecular  cohesion. 
Every  augmentation  of  a  liquid-surface  is  associated  with 
work  done  by  exterior  forces;  the  result  of  which  is  an 
augmentation  of  the  store  of  potential  energy  in  the  liquid 
and  the  quantity  of  this  potential  energy  depends  upon  the 
area  of  the  liquid-surface. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  p.  590.) 

NORMAL  PRESSURE  ON  A  LIQUID-SURFACE. 

The  forces  of  cohesion  which  act  upon  the  molecules  of 
the  superficial  layer  of  a  liquid  result  in  producing  a  certain 
normal  pressure  per  unit  area  upon  the  surface  of  the  liquid. 
The  magnitude  of  this  pressure  depends  upon  the  form  of 
the  surface.  If  K  is  its  magnitude  for  a  plane  surface,  the 
normal  pressure  is  greater  than  K  on  a  convex  surface  and 
less  than  K  on  a  concave  surface. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  p.  590.) 

FORM  ASSUMED  BY  A  LIQUID-MASS  UNDER  THE  INFLUENCE 
OF  SURFACE  TENSION  ALONE. 

A  liquid-mass  not  subjected  to  any  exterior  force  is  in 
equilibrium.  The  pressure  exerted  upon  it  by  forces  result- 
ing from  surface-tension  is  the  same  at  all  points  of  the 
surface  of  the  mass  and  the  surface  of  the  liquid  assumes 
at  all  points  the  same  mean  curvature. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  p.  601.) 


40  LAWS  OF  PHYSICAL  SCIENCE 

PRESSURE-DIFFERENCE  ON  THE  TWO  SIDES  OF  A  SOAP-FILM. 

The  pressure-difference  p  on  the  two  sides  of  a  soap 
fllm  is  given  by  the  relation, 


Here  T  is  the  surface-tension  of  the  film,  and  Rt,  R2  are 
the  two  radii  of  Principal  Curvature  of  the  surface  of  the 
liquid-film  at  the  point  considered.  The  pressure  on  the 
concave  side  always  exceeds  that  on  the  convex  side  of  the 
film.  In  the  formula  above  the  convention  must  be  made 
that  a  radius  of  Principal  Curvature  is  to  be  taken  positive 
or  negative  according  as  the  corresponding  center  of  curva- 
ture falls  on  the  side  of  the  surface  where  the  pressure  is 
greater  or  on  the  opposite  side.  For  the  case  of  a  spherical 
soap-film  R!  =  R2  and  thus  the  normally  acting  inward 
pressure  on  the  inside  of  the  film  exceeds  that  on  the  outside 
by  an  amount 

T 

where  R  is  the  radius  of  the  spherical  film. 

(Poynting  and  Thomson,  Properties  of  Matter,  pp.  144- 
146.  Also  consult  Chwolson,  Traite  de  Physique,  Vol.  I, 
Part  5,  Chap.  IV.) 

VERTICAL  DISTANCE  BETWEEN  TWO  ELEMENTS  OF  A 
LIQUID-SURFACE. 

If  two  elements  of  the  surface  of  an  extended  liquid- 
mass  resting  on  a  plane  surface  be  selected  such  that  one 
of  these  elements  is  horizontal  and  the  other  vertical,  then 
the  perpendicular  distance  between  the  two  elements  is 
equal  to  the  square  root  of  the  capillary  constant  of  the 
liquid.  Or, 

'2T 


where  k  is  the  capillary  constant,  T  the  surface-tension  and 
8  the  density  of  the  liquid. 


HYDROSTATICS,  HYDRODYNAMICS  AND  CAPILLARITY   41 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  pp.  628- 
630.) 

ACTION   ON    LIGHT    FLOATING    BODIES    OF   SURFACE-TENSION. 

Surface-tension  will  cause  an  attraction  between  two  float- 
ing bodies  both,  of  which  are  wet  if  the  curved  portions  of 
the  liquid  surrounding  each  of  them  intersect  ;  the  same  is 
true  if  both,  the  bodies  are  not  wet.  If  one  is  wet  and  the 
other  is  not  they  will  repel  each  other  if  the  two  curved 
portions  of  the  liquid  intersect. 

(For  an  account  of  various  capillary  phenomena  see 
Ganot's  Physics,  art.  136.) 

WATER-WAVES,  SPEED  OF. 

1.  In  water  where  the  depth  h  is  small  compared  with  the 
wave-length,  the  speed  of  propagation  V  of  the  wave- 
crest  is, 


where  g  =  the  acceleration  of  gravity. 
2.  In  deep  water  where  the  depth  is  large  compared  with 
the  wave-length  1,  the  speed  is, 


-A/; 


27T 

Thus  a  deep  sea-wave  378  feet  long  travels  with  a  speed  of 
44  feet  per  sec. 

(Crew,  General  Physics,  p.  188.  Also  Mathematical  and 
Physical  Papers,  by  Lord  Kelvin,  Vol.  Ill,  p.  519.) 

RIPPLES  ON  SURFACE  OF  LIQUIDS. 

Very  short  waves  (less  than  1.6  cm.),  or  ripples,  may 
be  considered  to  be  propagated  by  the  surface-tension  of 
the  liquid  alone.  Ripples  of  short  wave-length  travel  faster 
than  ripples  of  long  wave-length  which  is  just  the  reverse 
of  what  happens  with  ordinary  water-waves  propagated 
by  gravity. 

(For  a  general  discussion  of  waves  and  ripples,  consult 
Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  pp.  706-709.) 


42  LAWS  OF  PHYSICAL  SCIENCE 

RIPPLES,   SPEED    OF. 

1.  Neglecting  the  effect  of  gravity  upon  the  speed  of  short 

waves  or  ripples,  the  velocity  is, 

f1 

where  T  is  the  surface-tension  of  the  liquid,  />  its 
density  and  1  is  the  wave-length. 

2.  When  account  is  also  taken  of  the  acceleration  g  of 

gravity 


/a-T+i 

V    IP          2r 


(The  slowest  water-waves  are  found  to  have  a  length 
of  about  1.6  cm.  "Waves  shorter  than  this  are  called 
"ripples.") 

(Crew,  General  Physics,  pp.  191,  192.) 


' 


HI 

SOUND 


SOUND 

DEFINITIONS. 

1.  Sound  in  the  physical  sense  is  either  the  vibrations  of  its 

source,  or,  of  the  elastic  medium  surrounding  the 
source. 

2.  Noise  is  a  sound  resulting  from  irregular  and  practically 

unanalyzable  vibrations. 

3.  Musical  tones  are  distinguished  by: 

a.  their  force  determined  by  the  amplitude  of  the  vibra- 
tions, 

b  their  pitch  determined  by  the  frequency  of  the  vibra- 
tions, 

c.  their  quality  determined  by  the  harmonics  present. 

4.  A  periodic  motion  is  one  which  constantly  returns  to  the 

same  condition  after  equal  intervals  of  time. 

PROPAGATION   OF   SOUND. 

Sound  is  propagated  as  a  longitudinal  wave  in  any 
elastic  medium.  It  cannot  be  transmitted  through  vacuous 
space,  as  the  presence  of  an  elastic  medium  is  essential. 

(Ganot's  Physics,  arts.  224,  225.) 

INTENSITY  OF  SOUND,  (i) 

The  intensity  of  sound  is  inversely  as  the  square  of  the 
distance  of  the  sounding  body  from  the  ear.  That  is,  sound 
radiates  from  a  point  in  a  homogeneous  medium  so  that  the 
wave-front  is  spherical  in  form. 

(Ganot's  Physics,  art.  227.) 

45 


46  LAWS  OF  PHYSICAL  SCIENCE 

INTENSITY  OF  SOUND,     (a) 

The  intensity  of  sound  in  a  physical  sense  is  the  quantity 
of  energy  which  traverses  in  the  unit  of  time  the  unit  of 
area  normal  to  the  sonorous  ray.  In  this  sense  the  intensity 
of  a  sound  of  given  pitch  is  proportional  to  the  velocity  of 
the  sound,  to  the  density  of  the  medium  and  to  the  square 
of  the  amplitude.  In  a  formula  the  intensity  is, 

J  =  27r2  N2  a28V, 

where  N  =  frequency,  or  number  of  complete  vibrations  per 
second, 

a  =  the  amplitude  of  a  vibration, 
8  =  density  of  medium  and 
V  =  the  velocity  of  propagation. 
(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  7,  p.  905.) 

INTENSITY  OF  SOUND.    (3) 

The  intensity  of  sound  is  modified  by  the  motion  of  the 
atmosphere  and  the  direction  of  the  wind  and  is  strengthened 
by  the  neighborhood  of  a  sonorous  body. 

(Ganot's  Physics,  art.  227.) 

SOUND-INTENSITY  IN  TUBES. 

In  a  speaking-tube  the  intensity  of  sound  does  not  de- 
crease with  the  square  of  the  distance. 

According  to  experiments  carried  on  by  Regnault,  the 
distance  to  which  a  sound  will  carry  in  such  a  tube  is 
roughly  proportional  to  its  diameter. 

(Ganot's Physics,  art.  229.) 

EXPANSIONS  AND   CONTRACTIONS   IN   SOUND-TRANSMISSION 
ARE  ADIABATIC. 

In  transmission  of  sound-waves,  the  expansions  and  com- 
pressions of  the  medium  occur  so  rapidly  that  the  expansions 
and  compressions  are  adiabatic ;  namely,  they  occur  in  such 
manner  that  no  heat  is  gained  or  lost  to  the  volume  of  gas 
considered. 

(Consult  Ganot's  Physics,  arts.  231  and  507.) 


SOUND  47 

VELOCITY  OF  SOUND,  GENERAL  PRINCIPLE. 

The  velocity  of  sound  in  any  fluid  equals  the  velocity 
acquired  by  a  body  in  falling  through  one-half  the  height 
which  represents  the  rate  of  variation  of  the  pressure  of  the 
fluid  with  its  density  during  a  sudden  change  of  density. 
Thus  if  V  =  velocity  per  second,  g  =  acceleration  of  gravity, 
D  =  density  of  fluid  and  p  =  pressure,  in  the  gravitational 
system  of  units, 


dD 
(Rankine,  The  Steam  Engine,  p.  321.) 

VELOCITY    OF    SOUND    IN   AIR. 

In  dry  air  at  0°  C.  the  velocity  of  sound  is  331.7  meters, 
or  1088  feet,  per  second. 

This  velocity  increases  as  the  square  root  of  the  absolute 
temperature.  Or,  if  T  is  the  temperature  in  degrees  centi- 
grade, the  velocity  in  meters  per  second  at  temperature  T  is, 


(Consult  Ganot's  Physics,  art.  230.  Also  Chwolson, 
Traite  de  Physique,  Vol.  I,  Part  7,  p.  923,  and  Smithsonian 
Physical  Tables,  p.  102.) 

VELOCITY  OF  SOUND   AND  AIR-DENSITY. 

For  the  same  temperature,  the  velocity  of  sound  is  in- 
dependent of  the  density  of  the  air  and  consequently  of  the 
pressure  and  is  also  roughly  independent  of  the  intensity 
and  the  quality  of  the  sound. 

(Ganot's  Physics,  art.  230.) 


48  LAWS  OF  PHYSICAL  SCIENCE 

NEWTON'S   FORMULA   (modified  by  Laplace)   FOR   THE   VELOCITY   OF 
SOUND    IN    GASES. 

The  velocity  of  propagation  of  sound  in  a  gas  is  directly 
as  the  square  root  of  the  elasticity  of  the  gas  and  inversely 
as  the  square  root  of  its  density.  The  elasticity  exceeds  the 
isothermal  elasticity  P  (  measure  1  by  the  pressure)  by  an 
amount  y  which  is  the  ratio  of  tl  e  specific  heats  of  the  gas 
at  constant  pressure  and  at  coi  stant  volume.  Calling  p 
the  density  of  the  gas, 


For  air  y  =  1.41  and  the  velocity  of  sound  in  air  at  0°  C. 
is  331.7  meters  per  second. 

(Consult  Ganot's  Physics,  art.  231.  Also  Chwolson, 
Trait  e  de  Physique,  Vol.  1,  Part  7,  pp.  922-924.) 

DOPPLER'S    PRINCIPLE. 

When  a  sounding  body  approaches  the  ear  the  note 
perceived  is  higher  than  the  true  one,  but  if  the  source 
recedes  from  the  ear,  the  note  perceived  is  lower. 

If  n  =  frequency  of  sounding  body,  V  =  velocity  of  com- 
pressional  waves  and  v  =  velocity  of  body  toward  or  from  the 

ear,  the  pitch  heard  is      n'  =  —  —  ,     where  the  minus  sign 

VTv 

is  used  for  an  approaching  body  and  the  plus  sign  for  a 
receding  body. 

Doppler's  principle  may  be  extended  to  any  system  of 
waves  in  a  medium. 

(Ames,  Theory  of  Physics,  p.  160.) 

VELOCITY  OF  SOUND  IN  LIQUIDS  AND  SOLIDS. 

The  rule  of  Newton  that  the  velocity  of  propagation  of 


sound  equals  the     ~  I  elastlclt>'      holds  for  liquids  and  solids 
\    density 

as  well  as  for  gases. 


SOUND  49 

For  liquids  the  elasticity  is  the  ratio  of  the  pressure 
applied  to  the  compression  produced. 

For  solids  Young's  modulus  may  be  taken  as  the  value 
of  the  elasticity.  The  velocity,  as  in  the  case  of  gases, 
varies  with  the  temperature. 

(Ganot's  Physics,  arts.  234,  235.  Also  Chwolson,  Traite 
de  Physique,  Vol.  I,  Part  7,  pp.  929-933.) 

REFLECTION  OF   SOUND-WAVES:   ECHOES. 

1.  The  angle  of  reflection  is  equal  to  the  angle  of  incidence. 

2.  The  incident  sonorous  ray  and  the  reflected  ray  are  in 

the  same  plane  perpendicular  to  the  reflecting  surface. 

An  Echo  is  the  repetition  of  a  sound  caused  by  its 
reflection  from  some  surface  transverse  to  its  line  of 
propagation. 

(Ganot's  Physics,  arts.  236,  237.) 

CHANGE   OF  PHASE   AT  REFLECTION. 

When  waves  of  sound  pass  from  a  less  into  a  more 
dense  medium,  a  portion  of  the  energy  is  reflected  back 
from  the  bounding  surface  without  change  of  phase.  If 
the  waves  are  passing  from  a  more  into  a  less  dense  medium, 
the  reflected  wave  undergoes  a  change  in  phase  of  one-half 
wave-length  at  the  reflecting  surface. 

(Poynting  and  Thomson,  Sound,  pp.  104-108.) 

PRINCIPLE  OF  RESONANCE. 

One  vibrating  system  may  resonate,  or  be  set  into 
sympathetic  vibration  by  another  separate  vibrating  system 
when  their  natural  periods  of  vibration  are  nearly  equal. 
The  more  accurately  they  are  tuned  together  the  more 
marked  is  the  resonance. 

(Poynting  and  Thomson,  Sound,  pp.  58-62.  Also 
Kimball,  College  Physics,  pp.  204  et  seq.) 


50  LAWS  OF  PHYSICAL  SCIENCE 

REFRACTION   OF   SOUND,     (i) 

A  sound-wave  is  refracted  upon  passing  from  a  medium 
of  one  density  into  a  medium  of  a  different  density.  Sound 
may  be  deflected  with  a  prism  or  focused  with  a  lens. 
Sound  travels  poorly  against  the  wind,  because  its  wave- 
front  is  tilted  upward. 

(Ganot's  Physics,  art.  238.) 

REFRACTION   OF   SOUND.     (2) 

When  a  sound-wave  passes  obliquely  from  one  medium 
to  another  in  which  its  velocity  is  different,  its  direction 
of  propagation  is  changed.  The  laws  of  refraction  are : 

1.  The  normals  to  the  incident  and  refracted  wave-fronts 

and  to  the  plane-surface  all  lie  in  the  plane  of  incidence. 

2.  The  ratio  of  the  sine  of  the  angle  of  incidence  to  the 

sine  of  the  angle  of  refraction  is  constant  for  a  given 
form  of  matter  and  waves  of  definite  wave-number. 
It  is  entirely  independent  of  the  angle  of  incidence 

itself.    ^  =  ju    is  called  the  index  of  refraction, 
smr 

(Consult  Ames,  Theory  of  Physics,  pp.  424,  425.  Also 
Ganot's  Physics,  arts.  238,  546,  547.) 

INTERFERENCE  OF  SOUND. 

fcound-interference  can  occur  between  two  sonorous 
rays;  the  interference  being  determined  by  a  difference  in 
phase  between  the  vibrations  of  the  two  wave-trains. 

(Consult  Chwolson,  Traite  de  Physique,  Vol.  I,  Part  7, 
pp.  949,  950.) 

DIFFRACTION    OF   SOUND. 

Diffraction  phenomena  are  manifested  more  markedly 
with  long  than  with  short  sound-waves  and  the  length  of 
sound-waves  is  such  that  there  scarcely  exists  anything  of 
the  nature  of  a  sound-shadow. 


SOUND  51 

Lord  Rayleigh  has  experimentally  demonstrated  the 
phenomena  of  the  diffraction  of  sound. 

(Chwolson,  TraiU  de  Physique,  Vol.  I,  Part  7,  pp. 
954,  955.) 

VELOCITY  OF  A  TRANSVERSE  WAVE  ALONG  A  STRETCHED  STRING. 

The  velocity  of  propagation  of  a  transverse  disturbance 
along  a  perfectly  flexible  stretched  string  or  wire  is  given  by 


\m' 


Here  V  may  be  taken  as  the  velocity  in  cm.  per  second,  the 
tension  of  the  string  in  dynes  and  m  its  mass  in  grains  per 
cm.  of  its  length. 

(Poynting  and  Thomson,  Sound,  pp.  93-95.    Also,  Kim- 
ball,  College  Physics,  p.  216.) 

TRANSVERSE   VIBRATIONS   OF  A  CORD. 

1.  The  number  of  vibrations  per  second  made  by  a  cord 

under  a  given  tension  is  inversely  proportional  to  the 
length  of  the  vibrating  segment. 

2.  In  case  of  two  cords  of  equal  length,  and  equal  mass 

per  unit  length,  the  frequencies  are  proportional  to 
the  square  roots  of  the  tensions. 

3.  If  two  cords  have  equal  lengths  and  are  under  equal 

tensions,  their  frequencies  will  be  inversely  propor- 
tional to  the  square  roots  of  their  masses  per  unit 
length. 
Thus  in  a  formula  the  frequency  is, 


where  1  equals  length  of  cord  or  the  distance  between  two 
consecutive  nodes,  T  equals  the  tension  of  the  cord  and  m 
equals  the  mass  per  unit  length. 

(Kimball,  College  Physics,  p.  216.    Also  Ganot's  Physics, 
art.  268.) 


52  LAWS  OF  PHYSICAL  SCIENCE 

NODES    AND    LOOPS    IN    AN    ORGAN-PIPE. 

In  a  closed  organ-pipe  the  top  is  always  a  node  or  point 
of  no  vibration  and  the  pipe,  when  the  air  column  vibrates 
to  the  fundamental  note,  is  one-quarter  wave-length  long. 
In  the  case  of  an  open  organ-pipe  there  is  an  antinode  at 
each  end  and  the  pipe,  when  the  air  column  vibrates  to 
the  fundamental  note,  is  one-half  wave-length  long.  Clos- 
ing the  end  of  a  pipe  lowers  the  tone  one  octave. 

(Ganot's  Physics,  art.  275.) 

NUMBER  OF  VIBRATIONS  PRODUCED  BY  A  MUSICAL  PIPE. 

When  the  length  L  of  the  pipe  exceeds  12  times  its 
diameter;  for  open  pipes  the  frequency  is, 


and  for  closed  pipes 


(2p-l)y 
4L 


Here  p  is  any  whole  number,  as  1,  2,  3,  etc.,  and  V  is 
the  velocity  of  sound  in  air. 
(Ganot's  Physics,  art.  277.) 

LAW  OF  VIBRATION  OF  GEOMETRICALLY  SIMILAR  SYSTEMS. 

When  two  vibrating  systems  are  made  of  the  same  mate- 
rial and  are  geometrically  similar  but  of  different  size  their 
periods  of  vibration  are  in  the  same  ratio  as  their  linear 
dimensions. 

(Kimball,  College  Physics,  p.  230.) 

VIBRATION    OF    RODS    AND    PLATES. 

The  number  of  transverse  vibrations  made  in  a  given 
time  by  rods  and  thin  plates  of  the  same  material  is 
directly  as  their  thickness  and  inversely  as  the  square  of 
their  length. 

(Ganot's  Physics,  art.  283.) 


SOUND  «3 

VIBRATION  OF  PLATES. 

In  plates  of  the  same  kind  and  shape,  and  giving  the 
same  system  of  nodal  lines,  the  number  of  vibrations  in  a 
second  is  directly  as  the  thickness  of  the  plates  and  in- 
versely as  their  area. 

(Ganot's  Physics,  art.  284.) 

VIBRATION  OF  BELLS. 

Bells  may  be  considered  as  curved  plates.  They  do  not 
vibrate  as  a  whole,  but  for  the  fundamental  they  vibrate 
in  four  equal  parts,  these  parts  being  separated  by  nodal 
lines.  They  are  also  capable  of  vibrating  in  6,  8,  10  or 
12  parts,  producing  thus  a  series  of  overtones.  The  note  of 
a  bell  is  higher  in  proportion  as  the  surface  is  smaller  and 
the  substance  thicker. 

(Ganot's  Physics,  art.  284.) 

ACOUSTIC  ATTRACTION  AND   REPULSION. 

The  vibrations  of  an  elastic  medium  attract  bodies  which 
are  specifically  heavier  than  itself  and  repel  those  which  are 
specifically  lighter.  Thus  in  air  a  balloon  filled  with  car- 
bonic acid  gas  is  attracted  toward  the  opening  in  a  reso- 
nance-box on  which  is  a  vibrating  tuning  fork,  and  a  balloon 
filled  with  hydrogen  is  repelled. 

(Ganot's  Physics,  art.  292.) 

NUMERICAL  VALUE  OF  PRESSURE   OF   SOUND. 

Lord  Rayleigh  has  shown  that  acoustic  vibrations  when 
they  encounter  the  surface  of  a  body  must  exert  on  it  a 
pressure  p,  which  for  a  plane-wave  and  normal  incidence 
on  a  perfectly  reflecting  surface  is, 


where  e  is  the  quantity  of  incident  energy  in  unit  time  and 
V  is  the  velocity  of  the  sound. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  7,  p.  909.) 


54  LAWS  OF  PHYSICAL  SCIENCE 

LIMITS    OF    AUDIBILITY. 

The  limits  between  which  the  frequencies  of  vibrations 
are  audible  vary  considerably  with  different  persons  and 
the  results  of  different  investigators  vary  through  rather 
wide  limits.  However,  the  lower  limit  seems  to  be  between 
16  and  24  vibrations  per  second,  and  the  upper  limit  between 
30,000  and  41,000  per  second.  Much  depends  also  upon  the 
intensity  and  quality  of  the  sound  as  to  whether  or  not  it 
will  be  audible. 

(Ganot's  Physics,  art.  244.) 

AMPLITUDE    NECESSARY   TO   MAKE    SOUND-WAVES    AUDIBLE. 

It  is  found  tjiat  sound-waves  are  inaudible  if  the  ampli- 
tude of  the  sound-waves  is  less  than  about  8  X  10~8  cm. 
This  limit  is  smaller  as  the  pitch  is  higher. 

(Poynting  and  Thomson,  Sound,  pp.  118,  119.) 

COMBINATIONAL    TONES. 

When  two  pure  tones  are  sounded  simultaneously  there 
is  often  heard,  in  addition  to  these  two  tones,  two  others. 
One  of  these  has  a  pitch  of  frequency  equal  to  the  difference 
of  the  frequencies  of  the  original  tones,  and  the  other  has  a 
pitch  of  frequency  equal  to  their  sum. 

These  tones  are  called  Difference- Combinational  tones 
and  Summation- Combinational  tones.  They  have  been 
explained  by  Helmholtz. 

(Helmholtz,  Sensations  of  Tone.  Also  Encyclopedia 
Britannica,  10th  Ed.,  Vol.  XXV,  p.  56.) 

PURE    TONE. 

A  pure  tone  is  due  to  the  disturbances  sent  out  by  a 
body  vibrating  with  simple  harmonic  motion.  Fourier 
has  shown  that  any  periodic  disturbance  may  be  made  up 
of  the  resultant  disturbances  caused  by  a  number  of  simple 
harmonic  motions. 

(Consult  Helmholtz,  Sensations  of  Tone,  Chap.  1.) 


SOUND  *5 

LAW  OF  G.  S.   OHM. 

1  'The  human  ear  perceives  pendular  vibrations  alone  as 
simple  tones,  and  resolves  all  other  periodic  motions  of  the 
air  into  a  series  of  pendular  vibrations,  hearing  the  series 
of  simple  tones  which  correspond  with  these  simple  vibra- 
tions. " 

(Helmholtz.  Sensations  of  Tone,  p.  56.) 

RESULTS    OF    VON    HELMHOLTZ'S    RESEARCHES. 

Simple  sounds,  as  those  produced  by  a  tuning  fork  with 
a  resonance-box,  are  soft  and  agreeable,  without  roughness 
but  weak  and  in  deeper  tones,  dull.  Musical  tones  accom- 
panied by  a  series  of  harmonics,  say  up  to  the  sixth,  in 
moderate  strength  are  full  and  rich.  They  are  grander  and 
more  sonorous  than  simple  tones.  Such  tones  are  produced 
by  the  pianoforte. 

(These  tone-qualities  are  very  fully  discussed  in  Chaps. 
IV  and  V  of  Helmholtz 's  Sensations  of  Tone.) 

THE    OCTAVE. 

"A  musical  tone  which  is  an  octave  higher  than  another, 
makes  exactly  twice  as  many  vibrations  in  a  given  time  as 
the  latter." 

(Helmholtz,  Sensations  of  Tone,  p.  13.) 

EFFECT  ON  EAR  OF  A  SYSTEM  OF  SOUND-WAVES, 

' '  When  several  sonorous  bodies  in  the  surrounding  atmos- 
phere simultaneously  excite  different  systems  of  waves  of 
sound,  the  changes  of  density  of  the  air,  and  the  displace- 
ments and  velocities  of  the  particles  of  the  air  within  the 
passages  of  the  ear,  are  each  equal  to  the  (algebraical)  sum 
of  the  corresponding  changes  of  density,  displacements,  and 
velocities,  which  each  system  of  waves  would  have  separately 
produced,  if  it  had  acted  independently. " 

(Helmholtz,  Sensations  of  Tone,  p.  28.) 


56  LAWS  OF  PHYSICAL  SCIENCE 

ADDITION    OF   SIMPLE    VIBRATIONS. 

"  Any  given  regular  periodic  form  of  vibration  can 
always  be  produced  by  the  addition  of  simple  vibrations, 
having  pitch-numbers  which  are  once,  twice,  thrice,  four 
times,  etc.,  as  great  as  the  pitch-numbers  of  the  given 
motion. ' ' 

(Helmholtz,  Sensations  of  Tone,  p.  34.) 

THE  SUM  OF  PARTIAL  TONES. 

"Any  vibrational  motion  of  the  air  in  the  entrance  to 
the  ear,  corresponding  to  a  musical  tone,  may  be  always, 
and  for  each  case  only  in  one  single  way,  exhibited  as  the 
sum  of  a  number  of  simple  vibrational  motions,  correspond- 
ing to  the  partials  of  this  musical  tone/' 

(Helmholtz,  Sensations  of  Tone,  p.  34.) 

THE   PRINCIPLE   OF  MUSICAL    SCALES. 

In  the  major,  or  diatonic  scale,  the  frequencies  of  the 
notes  bear  the  following  ratios  to  that  of  the  key  note : 

do  re  mi  fa  sol  la  si  do 
1,  %.    %,  %,  %,    £%2 

This  scale  is  built  up  of  major  triads,  which  have  the 
pitch  relation  of  do,  mi,  sol,  or  the  frequency  relation  of 
4,  5,  6.  The  minor  scale  is  built  up  of  minor  triads,  or 
notes  with  the  frequency  ratios  5,  6,  7%.  The  notes  of 
this  scale  bear  to  the  key  note  the  relations 

1,  %:  %,  %,  %  %,  %  2. 
(Helmholtz,  Sensations  of  Tone,  p.  274.) 


SOUND  57 

HELMHOLTZ'S    THEORY    OF    CONSONANCE    AND    DISSONANCE. 

When  two  notes  are  sounded  simultaneously  they  pro- 
duce an  agreeable  sensation  in  proportion  as  their  fre- 
quencies form  a  simple  ratio.  Thus  the  octave,  1:2;  the 
fifth,  2 :  3  and  the  fourth,  3 : 4  are  the  most  consonant  com- 
binations, in  the  order  named.  A  ratio  8 :  9  or  7 : 11  would 
be  discordant.  Helmholtz  has  shown  that  these  facts  are 
due  to  the  absence  of  rapid  beats  in  the  case  of  consonant 
tones  and  their  presence  in  the  case  of  dissonant  tones. 

(Helmholtz,  Sensations  of  Tone,  pp,  228-330.) 


IV 

HEAT  AND  PHYSICAL  CHEMISTRY 


HEAT  AND  PHYSICAL  CHEMISTRY 

TEMPERATURE  (Definition). 

Temperature  is  a  condition  of  matter.  On  the  absolute 
thenno-dynamic  scale,  temperature  is  a  quantity  which  is 
proportional  to  the  mean  kinetic  energy  E  per  molecule 
of  the  molecules  of  an  "  ideal  gas,"  also  to  the  product  of 
the  volume  V  and  the  pressure  P  of  this  gas.  Thus, 

PV 

T=KE=  -if-,  where  K  and  R  are  constants. 
n 

(Consult  Maxwell,  Theory  of  Heat,  p.  51.  Also  Chwol- 
son,  Traite  de  Physique,  Vol.  Ill,  Part  9,  p.  7.  Also  article 
by  E.  F.  Northrup,  "High  Temperature  Investigation  and 
a  Study  of  Metallic  Conduction."  Journal  of  the  Franklin 
Institute,  June,  1915.) 

QUANTITY  OF  HEAT   (Definition). 

Quantity  of  Heat  is  the  total  kinetic  energy  of  the  mole- 
cules, or  ultimate  particles  of  a  body.  Thus  every  store 
of  heat-energy  is  expressed  by  the  formula, 


In  this  formula  the  unit  of  heat  energy  is  to  be  taken  the 
same  as  the  unit  of  mechanical  energy,  which  is  generally 
chosen  on  a  system  of  absolute  units  and  is  the  same  as  the 
unit  of  work.  By  m  is  to  be  understood  the  masses  of  the 
smallest  particles  of  the  body  which  are  moving  at  any  given 
instant  with  velocities  v,  different  in  general  for  each 
particle. 

(Chwolson,  Traite  de  Physique,  Vol.  Ill,  Part  9,  pp.  2 
and  18.) 

61 


62  LAWS  OF  PHYSICAL  SCIENCE 

TEMPERATURE    EQUILIBRIUM. 

If  two  bodies  A  and  B  are  in  temperature  equilibrium 
with  a  third  body  C,  then  A  and  B  will  be  in  temperature 
equilibrium  with  each  other.  It  does  not  follow  from  the 
above  that  A,  B,  and  C  contain  equal  quantities  of  heat, 
even  if  they  are  all  of  equal  mass  and  the  same  material. 
Thus  A  may  be  in  grams  of  water  at  0°  C.  and  B  in  grams 
of  ice  at  0°  C.  Then  A  and  B  will  be  in  temperature  equilib- 
rium with  C,  which  is  in  grams  of  ice  at  0  °  C.,  but  A  and 
B,  though  in  temperature  equilibrium  with  each  other,  con- 
tain different  quantities  of  heat  which  differ  by  about  80 
calories. 

(Preston,  Theory  of  Heat,  Chap.  I,  Sec.  II.    See  p.  20.) 

EQUALIZATION    OF    TEMPERATURE. 

When  two  bodies  A  and  B  are  placed  in  contact,  the 
temperature  of  the  one  body  being  higher  than  the  tem- 
perature of  the  other  body,  the  two  bodies  tend  toward 
equality  of  temperature.  The  equalization  occurs  or  tends 
to  occur  without  oscillations  of  heat  which  have  analogy 
with  the  oscillations  of  electricity,  as  observed  when  a  con- 
denser is  discharged.  The  progress  or  the  rate  of  equaliza- 
tion of  temperature  is  a  complex  phenomenon  which  has 
relations  with  specific  properties. 

NEWTON'S    LAW    OF    COOLING. 

The  rate  at  which  a  body  cools  is  proportional  to  the 
excess  of  its  temperature  above  the  walls  of  the  enclosure 

which   surround   it.     Thus,   log   -|r=-ait,    J  which  gives 

by  differentiation,  -~-  =  ae.  Here,  e  is  the  excess  of 
temperature  at  time  t,  E  the  initial  excess  of  temperature 
and  aj  and  a  are  constants. 

This  law  expresses  the  facts  only  approximately. 

(Poynting  and  Thomson,  Heat,  p.  245.  Also  Preston, 
Theory  of  Heat,  p.  528.) 


HEAT  AND  PHYSICAL  CHEMISTRY  63 

DULONG  AND  PETIT'S  CONCLUSIONS  ON  THE  VELOCITY  OF  COOLING. 

The  cooling  influence  of  gas  surrounding  a  body  is  not 
affected  by  the  nature  of  the  surface  of  the  body.  The 
nature  of  the  surface  is  effective  only  on  the  emissivity 
which  would  occur  if  the  body  were  in  a  vacuum. 

The  empirical  formula,  which  expresses  the  velocity  of 
cooling  V,  is 

V  =  k  (aP-aPo)  +  mpc  (0-00)  1.233. 

(For  discussion  of  principle  stated  and  interpretation 
of  formula,  see  Preston,  Theory  of  Heat,  pp.  530-540.) 

ABSOLUTE    (or  LORD   KELVIN'S)   SCALE   OF  TEMPERATURE. 

On  the  absolute,  thermodynamie  or  Lord  Kelvin's  scale 
of  temperature  any  two  temperatures  bear  to  each  other 
the  same  ratio  as  the  quantity  of  heat  taken  in  at  the  higher 
temperature  bears  to  the  quantity  of  heat  ejected  at  the 
lower  temperature  by  a  reversible  engine  working  between 
the  two  temperatures  as  source  and  condenser.  Thus, 

_TL  --  Qi 

T2         Q2 

The  efficiency  of  the  perfect  reversible  engine  is, 


T 

Here  the  Seat  quantities  are  expressed  by  Q  and  the  absolute 
temperatures  by  T. 

(Preston,  Theory  of  Heat,  p.  713.  Also  Rankine,  The 
Steam  Engine,  p.  343.) 

ABSOLUTE  ZERO  AND  ABSOLUTE  TEMPERATURE  (Definitions). 

The  absolute  zero  (-  273.10°  C.)  on  the  gas-scale  is  the 
temperature  at  which  an  ideal  gas  would  theoretically  exert 
no  pressure.  It  is  numerically  equal  to  the  reciprocal  of 
the  pressure-coefficient  av  of  the  gas  at  constant  volume. 


64  LAWS  OF  PHYSICAL  SCIENCE 

The  absolute  temperature  is  the  temperature  reckoned 
from  absolute  zero.  T  (degrees  absolute,  now  called  degrees 

Kelvin)    =  —  —  ht  (degrees  centigrade). 
av 

(Consult  Smithsonian  Physical  Tables,  p.  247.  Also 
Ganot's  Physics,  arts.  336,  337.  Also  paper  by  Arthur  L. 
Day  and  Robert  B.  Sosman,  "The  Nitrogen  Thermometer 
from  Zinc  to  Palladium/'  Amev*.  Jour,  of  Science  ,  Vol. 
XXIX,  Feb.,  1910.  See  pp.  100-102.  Also  Chwolson, 
Traite  de  Physique,  Vol.  Ill,  Part  9,  pp.  14-17.) 

CARNOT'S    THEOREM. 

All  reversible  heat  engines  working  between  two  given 
temperatures,  and  taking  in  and  ejecting  heat  at  the  same 
two  temperatures,  have  the  same  efficiencies.  This  efficiency 
is  greater  than  that  of  any  irreversible  engine  working 
between  the  same  two  temperatures. 

(Poynting  and  Thomson,  Heat,  pp.  262-264;  265,  266.) 

GAS-TEMPERATURE  SCALE. 

On  the  hydrogen  or  nitrogen  gas-thermometer  scale  the 
temperature  t,  in  degrees  centigrade,  is  given  by  the  relation 


where  P0  is  the  pressure  of  the  gas  at  0°  C.,  P100  its  pressure 
at  100°  C.  and  Pt  its  pressure  at  the  measured  temperature, 
the  volume  of  the  gas  being  maintained  constant. 

(Chwolson,  Traite  de  Physique,  Vol.  Ill,  Part  9,  p.  23. 
See  also  p.  17.) 


HEAT  AND  PHYSICAL  CHEMISTRY  65 

TEMPERATURE  BY  PLATINUM  RESISTANCE-THERMOMETER. 

Call  pt  =    Rt~R°  100  the   "  platinum     temperature,  " 
RIOO     R  o 

where  Et  is  the  resistance  of  the  thermometer  at  t  degrees 
and  E0  and  K100  its  resistance  at  0°  and  100°  on  the  centi- 
grade scale.  Then  the  difference  between  the  true  tem- 
perature and  the  platinum  temperature  is  given  by  the 
formula  of  Callendar, 


where  8  is  a  constant,  of  value  1.50  for  pure  and  greater  for 
impure  platinum. 

(Northrup,  Methods  of  Measuring  Electrical  Resistance  , 
p.  298.  Also,  Burgess  and  LeChatelier,  Measurement  of 
High  Temperatures,  p.  197.) 

EXPANSION    OF    BODIES    WITH    HEAT. 

Nearly  all  bodies  expand  when  they  receive  an  additional 
quantity  of  heat.  The  expansion  being  slight,  the  coefficient 
of  cubical  expansion  can  with  small  error  be  taken  equal 
to  three  times  the  coefficient  of  linear  expansion. 

For  small  ranges  of  temperature  the  expansion  is  very 
nearly  proportional  to  the  rise  in  temperature  of  the  body. 

(For  experimental  values,  consult  Smithsonian  Physical 
Tables,  pp.  232-235.) 

EXPANSION   COEFFICIENTS   OF  ANISOTROPIC   BODIES. 

1.  The  sum  of  the  coefficients  of  linear  expansion  along  any 

three  directions  mutually  at  right  angles  has  a  con- 
stant value  equal  to  the  sum  of  the  three  principal 
coefficients. 

2.  The  coefficient  of  cubical  expansion  for  an  anisotropic 

body  is  equal  to  the  sum  of  the  coefficients  of  linear 
expansion  along  three  directions  mutually  at  right 
angles. 

(Chwolson,  Traite  de  Physique,  Vol.  Ill,  Part  9,  pp. 
110,  111.) 


C6  LAWS  OF  PHYSICAL  SCIENCE 

EXPANSION    OF    LIQUIDS. 

Liquids  in  general  expand  when  they  receive  an  addi- 
tional quantity  of  heat,  but  water  between  0°  and  4°  C. 
contracts,  or  increases  in  density,  with  increase  in  tempera- 
ture or  heat  absorbed. 

The  real  or  absolute  expansion  of  a  liquid  is  the  actual 
increase  in  volume,  while  the  apparent  expansion  is  that 
which  is  observed  when  a  liquid  contained  in  a  vessel  is 
heated,  and  this  is  less  than  the  real  expansion,  because  of 
the  simultaneous  expansion  of  the  vessel  itself. 

(Ganot's  Physics,  arts.  322,  327,  331.) 

DULONG  AND   PETIT'S   LAW   OF   THERMAL    CAPACITY. 

For  simple  substances  the  atoms  all  have  (approxi- 
mately) the  same  thermal  capacity,  or  the  product  of  the 
specific  heat  by  the  atomic  weight  is  the  same  for  all 
elementary  substances. 

Regnault's  mean  value  of  this  constant  for  32  sub- 
stances is  6.38. 

(Preston,  Theory  of  Heat,  p.  294.  Also  Ganot's  Physics, 
art.  464.) 

NEUMANN'S    LAW. 

F.  E.  Neumann  has  found  that  the  product  of  the  molecu- 
lar weight  and  specific  heat  remains  constant  for  all  com- 
pounds belonging  to  the  same  general  formula  and  similarly 
constituted,  but  that  the  product  varies  from  one  series  to 
another. 

(Consult  Preston,  Theory  of  Heat,  p.  296.  Also  Ganot's 
Physics,  art.  465.) 

HEAT-FLOW:    LAW    OF   FLOW    FOR   STEADY    STATE. 

The  quantity  of  heat  Q  which  passes  through  a  homo- 
geneous solid  enclosed  between  two  parallel  infinite  planes 
at  a  distance  d  apart  is  expressed  by, 


HEAT  AND  PHYSICAL  CHEMISTRY  67 

where  BI  =  the  higher  temperature  of  the  one  plane, 
62  =  the  lower  temperature  of  the  other  plane, 
A  =  the  area  through  which  the  flow  is  reckoned, 
t  =  the  time  the  flow  is  measured  and 
K  =  a  constant. 

(Fourier,  The  Analytical  Theory  of  Heat,  Chap.  I,  Sec. 
IV.  For  the  steady  flow  of  heat  between  opposite  faces  of 
solids  with  certain  geometric  forms,  see  paper  by  Langmuir, 
Adams  and  Meikle  in  Trans,  of  the  Electrochem.  Soc.,  Vol. 
XXIV,  1913,  pp.  53-84.  Also  paper  by  B.  F.  Northrup, 
same  Vol.,  pp.  85-106.) 

GENERAL    EQUATION    FOR    HEAT-FLOW. 

The  general  differential  equation  for  heat-flow  is, 


which,  for  the  steady  state,  becomes 
«         " 


c~dt~ 


a 


dx2 
Here,  K  =  thermal  conductivity, 

c  =  thermal  capacity  per  unit  volume  =  specific  heat 
X  density. 

0  =  temperature  and 
t  =  time. 

(Preston,  Theory  of  Heat,  art.  312.    Also  Fourier,  The 
Analytical  Theory  of  Heat,  p.  112  et  seq.) 


68  LAWS  OF  PHYSICAL  SCIENCE 

STEADY  FLOW  OF  HEAT  FROM  A  POINT-SOURCE  IN  AN  INFINITE 
ISOTROPIC  MEDIUM. 

1.  The  isothermal  surfaces  are  concentric  spherical  shells 

with  the  point-source  as  center. 

2.  The  flow  of  heat  is  perpendicular  to  the  isothermal  sur- 

faces. 

3.  The  total  flow  of  heat  across  any  isothermal  surface  is 

the  same  as  that  across  any  other,  or  any  heat  that  is 
once  within  a  tube  of  flow  remains  in  it  forever. 

4.  The  flow  per  unit  area  through  any  cross-section  of  a 

tube  of  flow  varies  inversely  as  the  area  of  the  section, 
and  hence  inversely  as  the  square  of  the  distance  from 
the  point-source. 
(Preston,  Theory  of  Heat,  art.  313.) 

FLOW  OF  HEAT  IN  AN  INFINITE  CRYSTALLINE  MEDIUM. 

In  an  infinite  crystalline  medium,  if  heat  be  introduced 
at  a  single  point,  the  isothermal  surfaces,  when  the  steady 
state  is  reached,  will  be  a  system  of  concentric  and  similar 
ellipsoids,  the  axes  of  any  one  of  which  are  directly  pro- 
portional to  the  square  roots  of  the  three  principal  conduc- 
tivities of  the  crystalline  medium. 

(Preston,  Theory  of  Heat,  p.  675.) 

FIRST    LAW    OF    THERMODYNAMICS. 

When  work  is  done  (namely,  when  measurable  forces  act 
through  measurable  distances  or  measurable  electromotive 
forces  give  rise  to  measurable  currents  or  measurable  cur- 
rents pass  through  measurable  resistances,  etc. ) ,  there  is  an 
equivalence  between  the  work  so  done  and  the  heat  de- 
veloped. This  is  expressed  by  the  equation, 
W  — JH  +  w. 

Here  W  is  the  total  work  done  and  H  the  heat  developed. 
J  is  the  equivalent  of  the  work  done  in  producing  heat,  and 
w  is  a  quantity  to  express  the  processes  occurring  which 


HEAT  AND  PHYSICAL  CHEMISTRY  69 

cannot  be  measured  as  heat,  such'  as  the  production  of 

sound,  radiant  energy,  etc.     When  w  is  zero, 

W  (in  kilogram-meters)  =  426.9  X  H (in kilogram-calories). 

Or  (according  to  Smithsonian  Physical  Tables,  p.  237) 
W  (in  ergs)  =  4.181  X  107  H  (in,  20°  C.,  gram-calories). 

(Consult  Hering,  Conversion  Tables,  pp.  72  and  171. 
Also  Preston,  Theory  of  Heat,  art.  37.  For  a  precise  state- 
ment, see  Nernst,  Theoretical  Chemistry,  pp.  7-10.) 

SECOND    LAW    OF    THERMODYNAMICS. 

"Heat  can  never  pass  from  a  colder  to  a  warmer  body 
without  some  other  change,  connected  therewith,  occurring 
at  the  same  time."  (Clausius.) 

(Clausius  on  Heat,  p.  117.  Also  Preston,  Theory  of 
Heat,  p.  49.  For  a  scholarly  treatment  of  the  fundamental 
principles  of  thermodynamics,  consult  Chwolson,  Traite  de 
Physique,  Vol.  Ill,  Part  9,  Chap.  VIII,  pp.  409-550.) 

SECOND  LAW  OF  THERMODYNAMICS  (Rankine's  statement),  (i) 

' '  If  the  total  actual  heat  of  a  homogeneous  and  uniformly 
hot  substance  be  conceived  to  be  divided  into  any  number 
of  equal  parts,  the  effects  of  those  parts  in  causing  work  to 
be  performed  are  equal." 

(Rankine,  The  Steam  Engine,  p.  306.) 

SECOND  LAW  OF  THERMODYNAMICS  (Rankine's  statement),  (a) 

"If  the  absolute  temperature  of  any  uniformly  hot  sub- 
stance be  divided  into  any  number  of  equal  parts,  the  effects 
of  those  parts  in  causing  work  to  be  performed  are  equal." 

(Rankine,  The  Steam  Engine,  p,  307.) 


70  LAWS  OF  PHYSICAL  SCIENCE 

TRANSFORMATION   OF   ENERGY. 

1 1  The  effect  of  the  presence  in  a  substance,  of  a  quantity 
of  actual  energy,  in  causing  transformation  of  energy,  is 
the  sum  of  the  effects  of  all  its  parts. ' ' 

(This  general  law  was  first  enunciated  at  the  Phil.  Soc. 
of  Glasgow,  Jan.,  1853.  Consult  Rankine,  The  Steam  En- 
gine, p.  309.) 

DIFFERENCE  BETWEEN  ABSORBED  HEAT  AND  ENERGY. 

The  difference  between  the  whole  heat  absorbed,  and  the 
whole  expansive  energy  exerted  in  any  thermic  operation 
depends  on  the  initial  and  final  conditions  of  the  substance 
alone  in  respect  to  pressure  and  volume  and  not  on  the  inter- 
mediate process. 

(Rankine,  The  Steam  Engine,  p.  304.) 

CONVERTIBILITY    OF    ENERGY. 

"All  forms  of  energy  are  convertible.  The  total  energy 
of  any  substance  or  system  cannot  be  altered  by  the  mutual 
actions  of  its  parts. ' ' 

(Rankine,  The  Steam  Engine,  p.  299.) 

INTRINSIC    ENERGY. 

The  total  intrinsic  energy  of  a  body  or  system  of  bodies 
is  never  known.  When  bodies  mutually  react  it  is  only  the 
difference  of  the  energy  of  each  body  in  two  states  which  is 
considered.  If  a  body  has  less  energy  in  its  actual  than  in 
its  standard  state  the  expression  for  its  energy  is  negative. 

(Maxwell,  Theory  of  Heat,  p.  186.  For  a  general  treat- 
ment of  the  fundamental  energy  relations,  consult  Nernst, 
Theoretical  Chemistry,  pp.  7-10,  15-28.) 

ENTROPY. 

Entropy  is  a  mathematical  function  introduced  by 
Clausius.  The  entropy  S  of  a  substance  is  defined, 


HEAT  AND  PHYSICAL  CHEMISTRY  71 


where  dQ  is  an  element  of  a  quantity  of  heat  and  T  is  the 
absolute  temperature  of  the  element  of  the  body  which  con- 
tains the  element  of  heat  dQ.  In  any  change  of  condition 
of  a  body  the  change  in  its  entropy  is 


where  Sj  is  the  value  of  the  entropy  in  the  final  state  of  the 
body  and  S2  the  value  in  the  original  state  of  the  body,  and 
dQ  is  an  element  of  heat  gained  by  the  body  at  absolute 
temperature  T. 

(For  Clausius'  original  use  of  the  term  entropy  see 
Clausius  on  Heat,  p.  357.  Kankine  called  the  same  quantity 
the  " thermodynamic  function."  A  clear  explanation  of 
the  rather  elusive  meaning  of  the  term  entropy  is  to  be 
found  in  Maxwell,  Theory  of  Heat,  pp.  162,  187,  189.  Also 
Preston,  Theory  of  Heat,  pp.  724-726.) 

CLAUSIUS'  PRINCIPLE   OF  THE  INCREASE  OF  ENTROPY. 

(For  Clausius'  original  use  of  the  term  entropy  see 
total  entropy  of  the  universe.  The  entropy  of  the  universe, 
therefore,  tends  toward  a  maximum.  As  the  entropy  in- 
creases the  store  of  energy  capable  of  transformation  into 
useful  work  (available  energy)  diminishes,  approaching  a 
value  zero.  In  this  condition  the  entire  energy  of  the 
universe  will  be  in  the  form  of  heat  and  all  bodies  will  be 
at  the  same  temperature. 

The  above  statement  is  a  hypothetical  extension  to  the 
universe  of  a  principle  recognized  for  the  very  limited  por- 
tion which  can  be  studied. 

(Poynting  and  Thomson,  Heat,  p.  277  et  seq.) 


72  LAWS  OF  PHYSICAL  SCIENCE 

HEAT  PRODUCED  BY  RADIUM. 

Radioactive  matter  continually  evolves  heat.  Curie  and 
Laborde  conclude  from  experiments  that  1  gram  of  pure 
radium  emits  100  gram-calories  per  hour.  Thus  1  gram 
of  radium  emits  per  day  nearly  as  much  energy  as  will  dis- 
sociate 1  gram  of  water. 

(Rutherford,  Radio- Activity,  p.  159.) 

FUNDAMENTAL    LAWS    OF    GASES. 

Gases  obey  approximately  the  following  laws : 

1.  Boyle's  law,  that  for  a  constant  temperature  the  volume 

of  a  gas  diminishes  in  direct  proportion  to  the  pressure. 

2.  Gay-Lussac's  law,  that  the  volume  of  a  gas  at  constant 

pressure  increases  proportionally  with  the  absolute 
temperature. 

3.  Avogadro's  law,  that  equal  volumes  of  different  gases  at 

the  same  pressure  and  same  temperature  contain  equal 
numbers  of  molecules. 

4.  Dalton's  law,  that  the  pressure  of  a  mixture  of  several 

gases  in  a  given  space  is  equal  to  the  sum  of  the  pres- 
sures which  each  gas  would  exert  by  itself  if  confined 
in  that  space. 

5.  Joule's  law,  that  gases  in  expanding  do  no  interior  work. 

(For  the  laws  which  apply  to  a  perfect  gas,  consult 
Chwolson,  Traite  de  Physique,  Vol.  Ill,  Part  9,  Chap.  IX, 
p.  551  et  seq.  Also  Nernst,  Theoretical  Chemistry,  pp. 
38,  39.) 

BOYLE'S  (or  MARIOTTE'S)   LAW. 

The  temperature  being  constant,  the  volume  of  a  given 
quantity  of  gas  varies  inversely  as  the  pressure  which  it 
bears.  Thus,  PV  =  a  constant,  where  P  is  the  pressure  and 
V  the  volume  of  the  gas. 

This  law,  discovered  by  Boyle  in  1662  and  Mariotte  in 


HEAT  AND  PHYSICAL  CHEMISTRY  73 

France  in  1679,  is  only  approximately  true  for  actual  gases, 
and  then  only  for  low  or  medium  pressures. 

(Ganot's  Physics,  art.  181.  Also,  Chwolson,  Traite  de 
Physique,  Vol.  I,  Part  3,  p.  40  and  Part  4,  p.  422.) 

VARIATIONS  FROM  BOYLE'S  LAW. 

1.  No  actual  gas  obeys  Boyle's  law  rigorously.    The  diver- 

gence increases  with  the  pressure. 

2.  Hydrogen  is  less  compressible  than  Boyle 's  law  requires ; 

all  other  gases  are  more  compressible. 

3.  The  divergence  from  the  law  is  greater  for  the  easily 

liquefiable  gases,  such  as  carbonic  acid,  ammonia,  etc., 
than  for  the  gases  formerly  called  permanent  gases, 
oxygen,  nitrogen,  methane,  etc. 
(Ganot's  Physics,  art.  182.    For  a  precise  treatment  of 

Boyle's  and  other  gas-laws  and  their  variations  consult 

Nernst,  Theoretical  Chemistry,  pp.  37-53.) 

VAN  DER   WAALS'  FORMULA. 

Van  der  Waals  has  proposed  as  an  accurate  expression, 
relating  the  pressure  and  volume  of  a  gas  at  any  given 
temperature,  the  formula, 

f  p  -I 2  J  (  v  —  b  )  =  a  constant.     y. 

Here  p  is  the  pressure,  v  the  volume  and  a  and  b  are  con- 
stants which  differ  for  each  gas.  ^/ 

(Consult  Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4, 
pp.  439-441.  Also  Ganot's  Physics,  art.  183.) 


74  tLAWS  OF  PHYSICAL  SCIENCE 

GAY-LUSSAC'S    (or    CHARLES')    LAW. 

The  pressure  of  a  gas  being  maintained  constant  its 
volume  varies  directly  with  the  absolute  temperature.  If 
we  define  the  coefficient  of  expansion  a  of  a  gas  as  the  amount 
by  which  the  unit  of  volume  of  the  gas  at  0°  centigrade 
increases  when  the  temperature  is  raised  1°  C.,  the  pressure 
being  kept  constant,  then  the  law  gives, 

vt  =  v0(l  +  at), 

where  v0  is  the  volume  of  the  gas  at  0°  C.  and  vt  its  volume 
at  t°  C.  We  may  take  for  air  a  =  0.003665.  The  law  of 
Charles  holds  for  a  wide  variation  in  pressure. 

(Maxwell,  Theory  of  Heat,  p.  29.) 

VAN    DER    WAALS'    FORMULA  (combining   the   laws   of   Boyle   and 
Charles,   with   corrections): 


Here,  T  =  absolute  temperature, 
p  =  pressure, 
v  =  volume  and 
B  =  gas-constant. 
a  and  b  =  constants  which  differ  for  different  gases. 

When  for  the  unit  of  pressure  is  taken  a  column  of 
mercury  of  1  meter  and  for  the  unit  of  volume,  the  volume 
of  1  kilogram  of  gas  at  0°  C.  under  a  pressure  of  1  meter 
of  mercury,  from  Eegnault's  data: 

for  air  a  =  0.0037,    b  =  0.0026, 
for  C02  a  =  0.0115,    b  =  0.003, 
for  H2  a  =  0.0000,     b  —  0.00069. 
(Consult  Chwolson,  Trait  e  de  Physique,  Vol.  I,  Part  4, 
Chap.  II,  see  p,  441.) 

ADIABATIC   EXPANSION. 

The  law  of  expansion  of  a  perfect  gas,  without  receiving 
or  emitting  heat,  gives  for  the  relation  between  pressure  p 
and  volume  v, 


HEAT  AND  PHYSICAL  CHEMISTRY  75 

T£ 

p  =  -  ---  r,  where  K  is  a  constant 

For  air,  y  =  1.4025  and  for  steam  in  the  perfectly  gaseous 
state, 

y=  1.33. 

(Rankine,  The  Steam  Engine,  pp.  319,  320.  For  values 
of  y,  see  Smithsonian  Physical  Tables,  p.  243.) 

ADIABATIC   RELATIONS. 

The  adiabatic  relations  between  the  pressure  p  and  the 
absolute  temperature  T,  and  between  the  volume  v  and  the 
absolute  temperature  T,  are  given  by  the  two  relations, 

l-y 

Tp     y      =  a  constant,  and 

Q 

Tv?-1  =  a  constant,  where     7=  -^-  ,  the  ratio  of  the 

L/T 

specific  heat  at  constant  pressure  to  the  specific  heat  at  con- 
stant volume  of  the  gas. 

(Preston,  Theory  of  Heat,  p.  288.) 

VARIATION   OF  PRESSURE   WITH  VOLUME  IN  A  THERMALLY 
NON-CONDUCTING    VESSEL. 

The  rate  of  variation  of  the  pressure  with  the  volume, 
when  any  fluid  or  gas  is  enclosed  in  a  thermally  non- 
conducting vessel,  exceeds  the  rate  of  variation  when  the 
temperature  is  constant,  in  the  ratio  of  the  apparent  specific 
heat  of  the  fluid  at  constant  pressure  to  its  apparent  specific 
heat  at  constant  volume. 

Symbolically  expressed  (according  to  Rankine)  : 


dP   =  _  y     dT     ,  which  becomes  for  a  perfect  gas, 
dv  dv 

dT 

_dP_=_yJL. 
dv  v 

(Rankine,  The  Steam  Engine,  p.  320.) 


76  LAWS  OF  PHYSICAL  SCIENCE 

EQUIPARTITION  OF  ENERGY:  BOLTZMANN-MAXWELL  LAW. 

In  a  medium  (a  gas  is  usually  considered)  consisting 
of  particles  in  motion  the  distribution  of  energy,  throughout 
a  given  volume,  will  be  such  that,  on  the  average,  every 
mode  of  motion  of  its  particles  is  equally  favored,  or,  the 
kinetic  energy  is  uniformly  distributed  among  the  degrees 
of  freedom  of  the  particles. 

(Consult  Campbell,  Modern  Electrical  Theory,  p.  229. 
For  mathematical  treatment,  consult  Jeans,  The  Dynamical 
Theory  of  Gases,  pp.  67-69.) 

LAW    OF    AVOGADRO. 

When  two  gases  are  at  the  same  pressure  and  tempera- 
ture, the  number  of  molecules  in  unit  volume  is  the  same 
for  both. 

Let  N,  m,  u  be  the  number  of  molecules,  the  mass  of 
each  molecule  and  the  mean  velocity  of  each  molecule  of  the 
one  gas,  and  N±,  m1?  %  corresponding  quantities  for  the 
other  gas.  By  the  kinetic  theory  of  gases  %Nmu2  = 
^N^uJ  (where  volume,  pressure  and  temperature  are  the 
same),  but  %mu2  =  %  mu^  (by  equipartition  of  energy). 
Hence  the  law  that,  N  =  N^ 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  p.  491. 
Also  Nernst,  Theoretical  Chemistry,  pp.  200,  201.) 

LAWS  OF  THE  MIXTURE  OF  GASES  (DALTON'S  LAWS). 

1.  The  mixture  takes  place  rapidly  and  is  homogeneous; 

that  is,  each  portion  of  the  mixture  contains  the  two 
gases  in  the  same  proportion. 

2.  If  the  several  gases  and  the  mixture  have  the  same  tem- 

perature and  if  the  several  gases  and  the  mixture 
occupy  the  same  volume,  then  the  pressure  exerted  by 
the  mixture  will  equal  the  sum  of  the  pressures  ex- 
erted by  the  gases  severally. 
These  laws  are  applicable  to  mixtures  of  gases  and  vapors. 


HEAT  AND  PHYSICAL  CHEMISTRY  77 

(Preston,  Theory  of  Heat,  p.  71.  Also  Ganot's  Physics, 
art.  388.  Also  Chwolson,  Traite  de  Physique,  Vol.  I,  Part 
4,  p.  468.) 

JOULE'S   LAW. 

When  an  ideal  gas  expands  in  such  a  manner  as  not  to 
do  any  mechanical  work  its  temperature  does  not  change. 

Joule's  experimental  test  of  this  law  shows  that  no  in- 
ternal work  is  done  by  a  gas  during  expansion,  or  in  other 
words  no  molecular  attractions  have  to  be  overcome. 

Ordinary  gases  deviate  slightly  from  Joule's  law. 

(Preston,  Theory  of  Heat,  p.  286.  Also  Nernst,  Theo- 
retical Chemistry,  p.  42.) 

SPECIFIC  HEAT  OF  6ASES. 

It  is  concluded  from  experiments  by  Regnault  and  others 
that  a  gas  has  a  specific  heat  which  is  independent  of  pres- 
sure in  proportion  as  it  approaches  a  perfect  gas. 

(Preston,  Theory  of  Heat,  p.  '281.  For  discussion,  and 
description  of  experiments,  see  Chwolson,  Traite  de  Phy- 
sique, Vol.  Ill,  Part  9,  pp.  233-235.) 

SPECIFIC  HEAT  OF  A  GIVEN  VOLUME  OF  GAS. 

The  difference  between  the  specific  heats  under  constant 
pressure  and  under  constant  volume,  referred  to  the  unit 
of  volume,  is  the  same  for  all  perfect  gases  taken  at  the 
same  pressure  and  at  the  same  temperature. 

(Chwolson,  Traite  de  Physique,  Vol.  Ill,  Part  9,  p.  220.) 

LAW  OF  DELAROCHE  AND  BERARD. 

This  law  states  that  for  all  elementary  diatomic  gases 
approximately  in  the  perfect  state,  and  for  gaseous  com- 
pounds formed  without  condensation  and  approximately  in 
the  perfect  state,  the  product  of  the  molecular  weight  and 
the  specific  heat  at  constant  pressure  has  the  same  value. 

(New  Century  Dictionary  under  word  LAW.  Also  con- 
sult Nernst,  Theoretical  Chemistry,  p.  42.) 


78  .      LAWS  OF  PHYSICAL  SCIENCE 

INTERNAL  FRICTION  OF  A  GAS. 

It  follows  from  the  deductions  of  Maxwell  that  the  in- 
ternal friction  or  viscosity  of  any  gas  is  a  function  of  the 
absolute  temperature  but  is  independent  of  the  density  of 
the  gas. 

In  respect  to  the  density  the  statement  is  not  rigorously 
exact  for  actual  gases. 

(Chwolson,  Trait  e  de  Physique,  Vol.  I,  Part  4,  pp.  504- 
508.  Also  Poynting  and  Thomson,  Heat,  pp.  144-146.) 

THEOREM  OF  CORRESPONDING  STATES. 

If  the  pressure,  volume  and  temperature  of  any  gas  at 
its  critical  point,  be  chosen  for  the  unit  values  of  these 
quantities,  then  (assuming  its  accuracy  for  representing  the 
properties  of  a  particular  gas)  Van  der  Waals'  equation, 
with  uniform  values  of  the  constants,  will  apply  to  all  gases. 
In  other  words,  all  gases  exhibit  the  same  characteristics 
when  at  pressures  and  temperatures  which  are  proportional 
to  their  critical  pressures  and  temperatures. 

(Edser,  Heat  for  Advanced  Students,  pp.  312,  313.  Also 
consult  Nernst,  Theoretical  Chemistry,  pp.  219-226.) 

PRESSURE   OF  A   GRAM-MOLECULE   OF   GAS. 

The  molecular  weight  of  a  chemical  compound  expressed 
in  grams  is  called  a  gram-molecule  or  mol.  (A  gram-mole- 
cule or  mol  of  O2  is  32  grams,  of  H2  2  grams  and  of  H20 
18  grams.) 

The  pressure  exerted  by  one  gram-molecule  of  any  gas 
which  closely  obeys  the  gas-laws,  when  at  0°  C.  and  when 
occupying  a  volume  of  one  liter  is  22.412  atmospheres. 

(Nernst,  Theoretical  Chemistry,  pp.  40,  41.) 


HEAT  AND  PHYSICAL  CHEMISTRY  79 

THE    GAS-CONSTANT. 

For  any  gas  which  obeys  the  laws  of  an  ideal  gas 


nv  —      Q       T  —  T?T 

273  ' 

where  p0  and  v0  are  the  pressure  and  volume  respectively 
of  the  gas  at  0°  C.  and  p  and  v  its  pressure  and  volume 
respectively  at  any  absolute  temperature  T.  The  factor  R, 
called  the  gas-constant,  is  conditioned  only  by  the  units  of 
measurement  chosen  and  is  independent  of  the  number  of 
atoms  in  the  molecule  and  the  chemical  composition  of 
the  gas. 

If  p  is  measured  in  atmospheres,  v  in  liters  and  T  in 
centigrade  degrees  reckoned  from  absolute  zero,  the  value  of 
R  is  0.08204. 

(Nernst,  Theoretical  Chemistry,  p.  40.  Also  Smith- 
sonian Physical  Tables,  p.  342.) 

EQUATION  OF  CLAPEYRON. 

For  a  perfect  gas  pv  =  RT,  where  p  is  the  pressure  and 
v  the  volume  of  the  gas,  T  is  the  absolute  temperature  and 
R  is  a  constant.  The  constant  R  is  proportional  to  the  mass 
M  of  the  gas  and  for  equal  masses  inversely  proportional  to 
the  density  8.  Thus, 

M 
Roc—  , 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  pp.  437, 

438.) 


80  LAWS  OF  PHYSICAL  SCIENCE 

CONSTANT  OF  CLAPEYRON  AND  THE  WORK  OF  EXPANSION 
OF  A  GAS. 

The  constant  R  in  the  equation,  pv  =  RT,  is  numerically 
equal  to  the  work  of  the  expansion  of  the  gas  when  its 
temperature  is  raised  1°  C.  under  constant  exterior  pressure. 
Thus, 

R=^^CpJ,where7=SL 

~y  ^'v 

is  the  ratio  of  the  specific  heats  at  constant  pressure  and 
constant  volume  and  J  is  the  mechanical  equivalent  of  heat. 
(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  pp.  493- 
495.) 

FUNDAMENTAL    EQUATION   OF   THE    KINETIC    THEORY    OF   GASES. 

The  fundamental  equation  of  the  kinetic  theory  of 
gases  is, 

pv  =  1  Nmu2, 

where  v  is  the  volume  occupied  by  the  gas,  p  the  pressure 
of  the  gas,  N  the  number  of  molecules  contained  in  the 
volume  v,  m  the  mass  of  a  molecule  and  u2  the  mean  square 
velocity  of  translation  of  the  molecules. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  p.  484. 
Also  Preston,  Theory  of  Heat,  pp.  68-71.  See  serial 
article  by  Dr.  Saul  Dushmann  in  General  Electric  Review, 
Oct.,  Nov.,  Dec.,  1915,  on  "  The  Kinetic  Theory  of  Gases," 
Vol.  XVIII,  pp.  952-958,  1042-1049,  1159-1168.) 

PRESSURE   AND   ENERGY   OF   GAS. 

The  kinetic  theory  of  gases  states  that  the  pressure  of 
a  gas  is  equal  to  two-thirds  the  energy  of  translational 
motion  of  the  molecules  which  are  contained  in  the  unit  of 
volume  of  the  gas,  also  that  the  energy  of  translational 
motion  of  the  molecules  of  a  gas  is  proportional  to  the  abso- 
lute temperature  of  the  gas.  Thus,  for  the  unit  of  volume, 

p=-|-EiandRT=  _|_  ^ 
o  o 

where  p  is  the  pressure, 


HEAT  AND  PHYSICAL  CHEMISTRY  81 

T  the  absolute  temperature, 
B  a  constant  and 

E!  the  energy  of  translational  motion  in  the  unit  of 
volume. 

Important  relations  of  the  kinetic  theory  of  gases  are  i 

pv  =  |  Nmu2  =  |  MQ2  =  RT  =  |-E. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  pp.  489, 
490.) 

VELOCITIES    OF   GAS-MOLECULES. 

1.  The  velocity  of  the  molecules  of  a  given  gas  is  propor- 

tional to  the  square  root  of  the  absolute  temperature 
of  the  gas. 

2.  The  velocities  of  the  molecules  of  different  gases,  at  the 

same  temperature,  are  inversely  proportional  to  the 
square  roots  of  the  densities  of  these  gases.  Or  in 
a  formula, 


where  g  and  B0  are  constants,  T  absolute  temperature  and 
8  the  density  of  the  gas. 

(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  4,  p.  490.) 

MAXWELL'S    LAW    OF    MOLECULAR    VELOCITIES. 

The  components  of  molecular  velocity  (in  a  gas)  are 
distributed  among  the  molecules  according  to  the  same  law 
as  the  errors  are  distributed  among  the  observations  in  the 
theory  of  errors  of  observations. 

(Preston,  Theory  of  Heat,  pp.  71,  72.) 


82  LAWS  OF  PHYSICAL  SCIENCE 

WORK  PERFORMED  WHEN  TWO  GASES  MIX. 

When  two  gases  which  exhibit  no  chemical  interaction 
become  mixed  by  diffusion  of  each  into  the  other  no  work 
is  performed  if  the  volume  of  mixed  gases  remains  constant. 
If,  on  the  other  hand  (with  an  arrangement  which  may  be 
realized  experimentally),  the  volume  of  the  first  gas  in- 
creases from  Vi  to  V±  +  V2  and  the  volume  of  the  second  gas 
increases  from  V2  to  V2  +  V±,  then  the  first  gas  in  diffusing 
will  do  external  work: 

Wi  =  niRT  loge  —  L~^  >  and  the  second  gas  will  do  external 


work       W2  =  n2RTloge      l        2   - 

Here  T  is  the  absolute  temperature,  R  the  gas-constant, 
and  nx  and  n2  are  the  number  of  gram-molecules  concerned 
in  the  diffusion  of  the  first  and  second  gases  respectively. 

The  total  external  work  done  by  the  mixing  of  the  two 
gases  is, 


W  =  Wi  +  W2  =  RT   m  loge  +  n2  loge 


This  formula,  developed  by  Kayleigh  and  more  thor- 
oughly by  Boltzmann,  expresses  a  law  which  holds  good 
universally. 

(Nernst,  Theoretical  Chemistry,  pp.  96-100.) 

NUMBER   OF   MOLECULES   IN   A   GAS. 

It  is  deduced  from  theory  that,  in  a  cubic  centimeter 
of  air,  or  (according  to  the  law  of  Avogadro)  in  every  other 
gas,  there  are  contained  in  a  gram-molecule  of  the  gas  about 
4.5  X  1023  molecules. 

(Chwolson,  Trrite  de  Physique,  Vol.  I,  Part  4,  p.  511.) 


HEAT  AND  PHYSICAL  CHEMISTRY  83 

LAWS  OF  ABSORPTION  OF  GASES  BY  LIQUIDS:  HENRY'S  LAW. 

1.  For  the  same  gas,  the  same  liquid  and  the  same  tem- 

perature, the  weight  of  gas  absorbed  is  proportional  to 
the  pressure,  or,  at  all  pressures,  the  volume  dissolved 
is  the  same.  (Known  as  Henry's  law.)  The  volume 
absorbed  varies  with  the  gas.  Thus,  water  dissolves 
over  fifty  thousand  times  as  great  a  volume  of  ammonia 
as  of  nitrogen.  The  absorbing  power  also  varies  with 
the  liquid.  Thus  alcohol  absorbs  gases  better  than 
water. 

2.  The  quantity  of  gas  absorbed  decreases  with  increase  of 

temperature. 

3.  The  quantity  of  gas  which  a  liquid  can  dissolve  is  in- 

dependent of  the  nature  and  of  the  quantity  of  other 
gases  which  it  may  already  hold  in  solution. 
1  and  3  are  only  rigorously  exact  for  gases  which  are 
but  slightly  soluble  and  when  the  pressures  do  not  exceed  a 
few  atmospheres. 

(Ganot's  Physics,  art.  190.  Also  Walker,  Introduction 
to  Physical  Chemistry,  pp.  57,  58.) 

SOLUTION   IN   LIQUID   OF   MIXED    GASES:     DALTON'S    LAW. 

When  a  mixture  of  gases  dissolves  in  a  liquid,  each  com- 
ponent of  the  mixture  dissolves  proportionally  to  its  own 
partial  pressure;  or  when  a  liquid  acts  to  dissolve  mixed 
gases,  each  gas  dissolves  as  if  all  the  others  were  absent. 
(Known  as  Dalton's  law.) 

Both  Dalton  's  and  Henry 's  law  hold  well  only  when  the 
gases  are  slightly  soluble  and  the  pressures  do  not  exceed  a 
few  atmospheres.  The  divergencies  are  large  for  very  sol- 
uble gases  and  great  pressures. 

(Walker,  Introduction  to  Physical  Chemistry,  p.  58.) 


84  LAWS  OF  PHYSICAL  SCIENCE 

LAW    OF    PARTIAL    PRESSURE. 

In  a  mixture  of  liquids  the  vapor  emitted  by  the  liquids 
will  in  general  have  the  same  components  as  the  liquid 
mixture  remaining  behind.  The  ingredients  exert  partial 
pressures  the  sum  of  which  is  the  vapor-pressure  of  the 
mixture.  The  law  holds  good  universally  and  is  of  funda- 
mental importance,  that :  the  partial  pressure  of  each  com- 
ponent of  a  mixture  of  liquids  is  always  less  than  the  vapor- 
pressure  of  a  component  in  the  free  or  unmixed  state,  the 
temperature  being  the  same. 

(Nernst,  Theoretical  Chemistry,  pp.  105-107.) 

WORK  DONE  BY  EVAPORATION. 

A  liquid  which  evaporates  against  the  constant  external 
pressure  of  its  saturated  vapor  performs  external  work  and 
absorbs  heat.  The  external  work  performed  by  the  evapora- 
tion of  one  gram-molecule  of  any  simple  liquid  is  indepen- 
dent of  the  nature  of  the  liquid  and  is  directly  proportional 
to  the  absolute  temperature  T  at  which  the  evaporation 
takes  place.  Thus  the  external  work  performed  in  the 
evaporation  of  one  gram-molecule  of  any  liquid  is, 
W  =  p(V-  V)  =  RT  -  PV. 

Here  p  is  the  external  pressure  and  V  the  volume  of  the 
vapor.  V  is  the  volume  of  the  one  gram-molecule  of  the 
liquid  before  evaporation  started  and  R  is  the  gas-constant. 
When,  as  usual  V  is  negligible  and  p  is  in  atmospheres  and 
V  in  liters  we  have, 

W=0.0821T  liter-atmospheres. 
(Nernst,  Theoretical  Chemistry,  p.  57.) 

ABSORPTION  OF  GASES  BY  SOLIDS. 

The  surfaces  of  solids  by  exerting  an  attraction  on  the 
molecules  of  gases  become  covered  with  a  layer  of  condensed 
gas,  and  porous  solids,  which  present  an  extended  surface  to 


HEAT  AND  PHYSICAL  CHEMISTRY  85 

the  gas,  tend  to  absorb  it,  many  solids  absorbing  a  con- 
siderable quantity  of  gas.    This  absorption  takes  place  with- 
out any  chemical  change.     The  absorption  is  in  general 
greater  in  the  case  of  the  more  easily  liquefiable  gases. 
(Ganot's  Physics,  art.  194.) 

OCCLUSION    OF   GASES. 

At  a  high  temperature,  platinum  and  iron  allow  hydro- 
gen to  traverse  them  quite  readily.  Some  metals  will  absorb 
gases  when  cooling  and  give  them  off  when  heating.  This 
property  is  most  marked  with  palladium,  which  not  only 
absorbs  hydrogen  while  being  cooled  after  being  heated  but 
even  when  cold.  This  may  cause  a  palladium  wire  to 
lengthen  quite  perceptibly. 

(Ganot's  Physics,  art.  195.) 

LAW    OF    DIFFUSION    OF    GASES    (GRAHAM'S    LAW). 

The  quantity  of  a  gas  which  passes  through  a  porous 
diaphragm  in  a  given  time  is  inversely  as  the  square  root 
of  the  molecular  weight  of  the  gas  (or  its  density). 

(Ganot's  Physics,  art.  191.  Also  consult  Poynting  and 
Thomson,  Heat,  p.  327.) 

EFFUSION   OF  GASES. 

"When  gas  passes  through  a  small  aperture,  about  0.013 
mm.  in  diameter,  from  a  region  where  its  pressure  is  h 
(expressed  in  terms  of  the  height  of  a  column  of  the  gas 
which  would  exert  the  same  pressure  as  that  of  the  effluent 
gas),  into  a  region  where  its  pressure  is  h'  the  velocity  of 
efflux,  or  the  rate  of  effusion,  is, 


where  g  =  the  acceleration  of  gravity. 

Or  we  can  say:  the  velocities  of  efflux,  or  the  rates  of 
effusion  of  various  gases,  are  inversely  as  the  square  roots 
of  their  densities. 

(Ganot's  Physics,  art.  192.) 


86  LAWS  OF  PHYSICAL  SCIENCE 

BOILING. 

1.  The  temperature  of  ebullition,  or  the  boiling-point,  in- 

creases with  the  pressure. 

2.  For  a  given  pressure  boiling  begins  at  a  certain  tem- 

perature, which  varies  for  different  liquids,  but  which, 
for  equal  pressures,  is  always  the  same  in  the  same 
liquid. 

3.  Whatever  be  the  rate  of  input  of  heat  into  the  liquid,  as 

soon  as  boiling  begins  the  temperature  of  the  liquid 

remains  stationary. 

(Ganot's  Physics,  art.  366.  For  cases  of  " Superheat- 
ing "  see  Preston,  Theory  of  Heat,  p.  360.  Also  consult 
Nernst,  Theoretical  Chemistry,  pp.  63,  64.) 

BOILING  AND  VOLATILIZATION. 

If  a  substance  is  liquid  at  a  temperature  at  which  the 
pressure  of  its  vapor  equals  the  pressure  on  its  surface,  the 
substance  will  liquefy  and  boil,  but,  if  the  substance  is 
solid  at  a  temperature  at  which  its  vapor  has  the  pressure 
of  the  pressure  on  its  surface,  the  substance  changes  directly 
from  a  solid  to  a  vapor ;  namely,  it  will  volatilize  or  sublime. 

Water  is  an  example  of  the  first,  arsenic  and  carbon  are 
examples  of  the  second. 

(See  Preston,  Theory  of  Heat,  p.  370.  Also  Walker, 
Introduction  to  Physical  Chemistry,  p.  82.  Also  Nernst, 
Theoretical  Chemistry,  pp.  70,  476.) 

LATENT  HEAT  OF  VAPORIZATION:  TROUTON'S  LAW. 

For  different  liquids  the  latent  heat  of  vaporization  mul- 
tiplied by  the  molecular  weight  is  approximately  propor- 
tional to  the  absolute  temperature  at  which  vaporization 
occurs;  or  the  molecular  latent  heat  is  approximately  pro- 
portional to  the  absolute  temperature. 

Thus  calling  w  the  molecular  weight  of  the  vapor,  L  the 
latent  heat  of  vaporization  of  the  liquid  and  T  the  absolute 
temperature, 


HEAT  AND  PHYSICAL  CHEMISTRY  87 

wL 

— ~ —  =  a  constant. 

(Preston,  Theory  of  Heat,  p.  391.  Also  consult  Nernst, 
Theoretical  Chemistry,  pp.  272-274.) 

VAPOR-PRESSURE    IN    COMMUNICATING    VESSELS    AT    DIFFERENT 
TEMPERATURES. 

"When  two  vessels  containing  the  same  liquid,  but  at 
different  temperatures,  are  connected,  the  pressure  is  identi- 
cal in  both  vessels,  and  is  the  same  as  that  corresponding 
to  the  lower  temperature. ' ' 

The  liquid  distils  from  the  vessel  at  higher  temperature 
to  the  vessel  at  lower  temperature. 

(Ganot's  Physics,  art.  364.) 

VAPOR-PRESSURE   OF   MIXED   LIQUIDS. 

1.  Liquids  which  do  not  mix :  the  vapor-pressure  equals  the 

sum  of  the  vapor-pressures  of  the  constituents. 

2.  Liquids  which  mix  partially:  the  vapor-pressure  is  less 

than  that  of  the  sum  of  the  pressures  of  the.  constituents. 

3.  Liquids  which  mix  in  all  proportions:  the  diminution 

of  vapor-pressure  is  still  greater. 
(Preston,  Theory  of  Heat,  p.  406.) 

VAPOR  FORMATION  IN  A  VACUUM. 

If  any  simple  volatile  liquid  is  brought  into  a  vacuous 
space  the  liquid  evaporates  with  great  rapidity  and  evapor- 
ation continues  until  the  vapor  formed  exerts  a  certain 
definite  maximum  pressure.  If,  when  this  pressure  is 
reached,  some  liquid  still  remains  in  the  space  originally 
vacuous,  the  maximum  pressure  obtained  will  be  the  so- 
called  vapor-pressure  of  the  liquid  at  the  particular  temper- 
ature of  the  experiment.  When  the  temperature  increases 
the  vapor-pressure  increases,  and  usually  very  rapidly. 

(Nernst,  Theoretical  Chemistry,  pp.  56, 57.  Also  Ganot's 
Physics,  art.  355.) 


88  LAWS  OF  PHYSICAL  SCIENCE 

i 

CONDENSATION  OF  SATURATED  VAPOR. 

If  air  be  perfectly  free  from  dust  particles  it  may  be 
considerably  supersaturated  without  condensation  in  the 
form  of  a  cloud. 

If  air  be  ionized  condensation  is  accelerated,  the  negative 
ions  being  more  effective  nuclei  than  the  positive  ions.  This 
fact  has  been  utilized  in  counting  the  number  of  ions  in  a  gas. 

(Poynting  and  Thomson,  Heat,  pp.  168-172.  Also 
Thomson,  Conduction  of  Electricity  Through  Gases,  pp. 
163-187.) 

VAPOR-PRESSURE. 

The  pressure  of  a  vapor  in  contact  with  its  own  liquid 
depends  only  upon  the  temperature  and  is  independent  of 
the  relative  proportions  of  the  liquid  and  vapor. 

The  pressure  is  some  function  of  the  temperature  but 
not  a  linear  function. 

"When  the  pressure  upon  the  surface  of  a  liquid  is  due 
to  the  atmosphere,  the  liquid  will  boil  at  the  moment  when 
its  vapor-pressure  just  exceeds  that  of  the  atmosphere.  A 
diminution  of  1°  C.,  in  the  boiling-point  of  water,  corre- 
sponds to  an  ascent  of  about  1080  feet. 

(Preston,  Theory  of  Heat,  pp.  148  and  408.) 

LAW  OF  MASS-ACTION  (Credited  to  Guldberg  and  Waage). 

As  stated  by  the  authors  of  the  law,  the  rate  of  chemical 
action  is  proportional  to  the  active  mass  of  each  of  the  re- 
acting substances.  It  may  be  thus  stated:  when  any  sub- 
stance in  solution  enters  into  a  chemical  reaction,  the  amount 
of  reaction  in  the  unit  of  time  is  proportional  to  the  active 
mass  of  the  substance,  namely,  to  the  number  of  gram- 
molecules  of  the  substance  contained  in  unit  volume  of  the 
solution. 


HEAT  AND  PHYSICAL  CHEMISTRY  89 

(See  Walker,  Introduction  to  Physical  Chemistry,  p. 
277  et  seq.  Also  consult  Nernst,  Theoretical  Chemistry,  pp. 
443-446,  for  an  analytical  treatment  of  this  very  funda- 
mental law  of  chemical  kinetics.) 

LAW    OF   RELATIVE    PROPORTIONS    IN    EQUILIBRIUM. 

The  condition  of  physical  or  chemical  equilibrium  in  a 
heterogeneous  system  is  independent  of  the  relative  mass 
of  each  phase  present  in  the  system. 

Thus,  at  a  given  pressure  and  temperature  the  physical 
equilibrium  between  water  and  its  vapor  is  undisturbed  by 
an  increase  in  the  mass  of  either  phase;  and  the  chemical 
equilibrium  between  CaCO3,CaO  and  CO2  is  undisturbed 
by  a  change  in  the  quantity  by  weight  of  any  of  the 
substances  enumerated. 

(Nernst,  Theoretical  Chemistry,  p.  471  et  seq.) 

LAW    OF    DISTRIBUTION,    AMONG    SEVERAL    MOLECULAR    SPECIES. 

When  several  molecular  species  [as  acetic  acid,  giving 
in  both  vapor  form  and  in  solution  the  single  molecules 
CH3C03H  and  the  double  molecules  (GH3C03H)2]  evapo- 
rate at  constant  temperature  from  a  common  solvent  (as 
benzene)  into  a  fixed  vapor-space  the  ratio  of  the  concentra- 
tion in  the  vapor-space,  of  any  one  molecular  species,  to  its 
concentration  in  the  solvent  is  constant. 

This  distribution  in  the  quantity  of  a  molecular  species 
between  the  solvent  and  the  vapor-space  is  independent  of 
the  presence  of  other  molecular  species,  even  when  these 
latter  are  chemically  reactive  with  the  former. 

(Nernst,  Theoretical  Chemistry,  p.  491.  For  application 
of  the  ' '  distribution  law ' '  to  dilute  solutions  see  Washburn, 
Principles  of  Physical  Chemistry,  pp.  148-150.) 


90  LAWS  OF  PHYSICAL  SCIENCE 

THE    LAW    OF    HESS. 

The  total  quantity  of  heat,  disengaged  in  the  passage  of 
a  group  A  of  substances  to  a  group  B,  is  independent  of 
the  nature  of  this  passage,  namely,  of  the  character  and  of 
the  order  of  intermediate  reactions,  provided  the  physical 
state  (in  the  wide  sense  of  this  word)  of  the  groups  A  and 
B  is  the  same  in  all  cases. 

This  law  is  a  basic  principle  of  thermochemistry. 

(Consult  Chwolson,  Trait e  de  Physique,  Vol.  Ill,  Part 
9,  p.  284.) 

EFFECT  OF  TEMPERATURE  ON  BALANCED  CHEMICAL  ACTION. 

If  a  direct  chemical  action  gives  out  a  certain  quantity 
of  heat  per  gram-molecule  transformed,  the  reverse  reaction 
will  absorb  an  exactly  equal  quantity  of  heat;  and  rise  of 
temperature  always  affects  chemical  equilibrium  in  such  a 
manner  that  the  displacement  of  the  point  of  equilibrium 
takes  place  in  the  direction  which  will  determine  absorption 
of  heat. 

(See  Walker,  Introduction  to  Physical  Chemistry,  p. 
293.  Also  Nernst,  Theoretical  Chemistry,  pp.  673-676. 
Note  statement  of  LeChatelier  on  p.  676.) 

PROGRESS    OF    CHEMICAL    DECOMPOSITION. 

The  phenomena  of  chemical  decomposition  of  a  body  in 
a  confined  space  go  on,  if  one  of  the  elements  of  the  decom- 
position is  gaseous,  until  a  certain  pressure  is  attained  when, 
for  a  particular  temperature,  the  decomposition  ceases. 

Deville  used  the  word  ' '  Dissociation ' '  which  has  analogy 
under  the  above  conditions  with  vaporization  of  liquids. 

(Roscoe  and  Schorlemmer,  Treatise  on  Chemistry,  Vol. 
II,  pp.  129-132.) 


HEAT  AND  PHYSICAL  CHEMISTRY  91 

CRITICAL    TEMPERATURE. 

There  is  for  every  substance  a  critical  temperature  above 
which  the  substance  cannot  be  liquefied  with  pressure;  or 
there  is  for  each  gas  a  particular  or  critical  temperature  to 
which  the  gas  must  be  cooled  in  order  to  liquefy  it  with  any 
pressure. 

(Preston,  Theory  of  Heat,  p.  450  et  seq.  See  C02  curves, 
pp.  490,  491.  Also  consult  Nernst,  Theoretical  Chemistry, 
pp.  64-66.) 

MOLECULAR    SURFACE-ENERGY:    LAW    OF    EOTVOS. 

According  to  Eotvos,  the  work  required  to  form  the 
surface  of  a  spherical  gram-molecule  of  a  liquid  varies  with 
the  temperature  in  the  same  manner  for  all  liquids.  This 
law  is  stated  in  the  expression, 

^2/3  =  k(T  -  T0), 

where  v  is  the  volume  occupied  by  one  gram-molecule  of 
a  liquid  and  7  is  its  surface  tension.  T0  is  a  temperature 
taken  not  far  from  the  critical,  T  the  temperature  of  the 
liquid,  and  k  is  a  constant,  independent  of  the  nature  of 
the  liquid.  The  quantity  yv2/3  is  proportional  to  the 
molecular  surface-energy  of  a  sphere  formed  from  one  gram- 
molecule  of  the  liquid  in  question. 

The  law  of  Eotvos  has  importance  in  determining  the 
molecular  weight  of  liquids. 

(Nernst,  Theoretical  Chemistry,  pp.  275-277.) 

CURIE'S   LAW. 

In  paramagnetic  substances  the  magnetic  susceptibility, 
or  the  ratio  of  the  intensity  of  magnetization  to  the 
magnetizing  force,  is  inversely  proportional  to  the  absolute 
temperature. 

(Richardson,  The  Electron  Theory  of  Matter,  p.  378.) 


92  LAWS  OF  PHYSICAL  SCIENCE 

LAWS   OF  DIFFUSION  IN  LIQUIDS. 

1.  The  rate  at  which  the  diffusion  of  any  substance  goes  on 

is  proportional  to  the  rate  of  variation  of  the  strength 
of  that  substance  in  the  fluid  as  measured  along  the 
line  in  which  the  diffusion  takes  place. 

2.  The  rate  of  diffusion  varies  with  the  temperature. 

The  law  of  diffusion  of  matter  has  exactly  the  same 
form  as  that  of  the  diffusion  of  heat  by  conduction. 

(See  Maxwell,  Theory  of  Heat,  Chap.  XIX,  and  p.  276. 
Also  consult  Ganot's  Physics,  art.  140.  Also  Nernst,  Theo- 
retical Chemistry,  p.  151  et  seq:) 

OSMOSE,  OSMOSIS  OR  DIOSMOSE. 

When  two  liquids  which  will  mix  are  separated  only  by  a 
porous  membrane  there  is  a  movement  of  the  liquids  in  both 
directions  through  the  membrane.  The  greater  movement 
is  usually  from  the  less  dense  to  the  more  dense  liquid  so 
as  to  cause  the  level  of  the  more  dense  liquid  to  rise  above 
that  of  the  less  dense.  This  action  increases  with  the  tem- 
perature and  is  proportional  to  the  strength  of  the  solution. 

(Ganot's  Physics,  art.  139.  Also  Chwolson,  Traite  de 
Physique,  Vol.  I,  Part  5,  pp.  661-668.  Also  Nernst,  Theo- 
retical Chemistry,  pp.  125-127.) 

OSMOTIC    PRESSURE    AN    ANALOGUE    OF    GAS-PRESSURE. 

A  principal  feature  in  the  analogy  between  a  dissolved 
substance  and  a  gas  consists  in  the  correspondence  between 
the  energy  content  of  each,  this  energy  content  being  in- 
dependent, at  any  fixed  temperature,  of  the  volume  occu- 
pied by  a  given  mass  of  either.  Thus,  the  osmotic  pressure 
of  a  dissolved  substance  is  exactly  the  same  as  the  gas- 
pressure  which  would  be  exerted  if  the  solvent  were  removed 
and  the  dissolved  substance  in  gaseous  form  were  left  behind 
to  occupy  the  same  volume  at  the  same  temperature.  Thus 
the  gas-law  for  a  dissolved  substance  may  be  written 
PV  =  ET  ==  0.0821T  liter-atmospheres. 


HEAT  AND  PHYSICAL  CHEMISTRY  93 

Here  P  denotes  the  osmotic  pressure  in  atmospheres  of 
a  solution  which  contains  one  gram-molecule  of  the  substance 
dissolved  in  V  liters  of  solvent,  and  T  the  absolute  tempera- 
ture in  degrees  centigrade. 

(Nernst,  Theoretical  Chemistry,  p.  144.) 

RAOULT'S  LAW  ON  THE  LOWERING  OF  VAPOR-PRESSURE. 

"The  relative  lowering  of  vapor-pressure  experienced 
by  a  solvent  on  dissolving  a  foreign  substance  is  equal  to 
the  quotient  obtained  by  dividing  the  number  of  dissolved 
molecules  n,  by  the  number  of  molecules  N,  of  the  solvent. ' ' 
Thus  in  symbols 

P-P'  _..n  . 

p'        "   N 
(Nernst,  Theoretical  Chemistry,  pp.  144, 145.  See  also  p. 

263  et  seq.) 

GAS-LAWS    APPLIED    TO    SOLUTIONS. 

1.  The  osmotic  pressure  is,  at  constant  temperature,  pro- 

portional to  the  concentration  of  the  solution,  or 
inversely  proportional  to  the  volume  occupied  by  a 
given  quantity  of  the  dissolved  substance  (analogue  of 
Boyle 's  law) . 

2.  The  osmotic  pressure  is  proportional  to  the  absolute  tem- 

perature (analogue  of  Gay-Lussac's  or  Charles'  law). 

3.  Equal  volumes  of  isotonic   solutions — solutions   which 

exercise  the  same  osmotic  pressure — when  under  the 
some  pressure  and  at  the  same  temperature  contain 
the  same  number  of  molecules.  This  number  of  mole- 
cules is  the  same  as  that  of  a  gas  under  like  conditions 
of  volume,  pressure  and  temperature  (analogue  of 
Avogadro's  law) . 
(Chwolson,  Traite  de  Physique,  Vol.  I,  Part  5,  p.  666. 

See  also  Walker,  Introduction  to  Physical  Chemistry,  p. 

184.    Also  Nernst,  Theoretical  Chemistry,  p.  141  et  seq.) 


94  LAWS  OF  PHYSICAL  SCIENCE 

LAW  OF  KOHLRAUSCH  FOR  DILUTE  SALT-SOLUTIONS. 

The  molecular  conductivity  of  a  solution  (namely,  its 
electrical  conductivity  times  the  volume  of  the  solution 
which  contains  one  gram-molecule  of  the  dissolved  sub- 
stance) is  independent  of  the  concentration  of  the  solution 
for  very  dilute  salt-solutions.  This  constant  value  of  the 
molecular  conductivity  of  an  electrolyte,  at  infinite  dilution, 
is  the  sum  of  two  numbers,  one  of  which  depends  solely  upon 
the  speed  of  migration  of  the  positive  ions  and  one  solely 
upon  the  speed  of  migration  of  the  negative  ions. 

(Walker,  Introduction  to  Physical  Chemistry,  pp.  251, 
256.  Also  Nernst,  Theoretical  Chemistry,  pp.  365-367.) 

ADDITIVE   PROPERTY   OF   DILUTE    SOLUTIONS. 

In  a  sufficiently  dilute  electrolytic  solution  there  is  a 
complete  dissociation  into  ions  of  the  dissolved  substance 
and  it  is  a  fundamental  law  that:  "the  properties  of  a  salt- 
solution  are  composed  additively  of  the  properties  of  the 
free  ions." 

The  law,  to  hold  true,  presupposes  complete  dissociation. 

(Nernst,  Theoretical  Chemistry,  p.  384.) 

SIMILARITIES  IN  BEHAVIOR  OF  IONS  AND  MOLECULES. 

It  may  be  stated,  as  a  theorem,  that:  the  ions  of  dis- 
sociated substances  in  solution  exhibit  all  the  properties  of 
neutral  molecules  and  some  additional  properties  due  solely 
to  the  electrical  charge  of  the  former.  Thus  ions,  like 
molecules,  diffuse,  exert  pressure,  distribute  according  to 
the  law  of  equipartition  of  energy,  show  additive  properties, 
etc.,  and  in  addition,  in  virtue  of  their  electric  charge, 
transport  electricity. 

(Nernst,  Theoretical  Chemistry,  pp.  392,  393.) 


HEAT  AND  PHYSICAL  CHEMISTRY 
OSTWALD'S    LAW    OP    MOLECULAR    CONDUCTIVITY. 

This  law  is  expressed  by  the  formula, 

m2 

-=  k,  a  constant. 


(1  -m)v 

k  is  called  the  dissociation  constant,  v  is  the  dilution  and  m 
represents  the  degree  of  ionization  and  is  equal  to 

Hv 

C1 

where     uv     is  the  molecular  conductivity  at  dilution  v  and 

H&    is  the  molecular  conductivity  at  infinite  dilution. 

(Walker,  Introduction  to  Physical  Chemistry,  Chap. 
XXV,  see  pp.  262-265.) 

FORMATION  OF  "  HYDRION." 

Aqueous  solutions  of  various  acids  possess  one  property 
in  common,  on  the  dissociation  theory:  they  form  "hy- 
drion, ' '  namely,  yield  hydrogen  atoms  in  the  ionic  state. 

The  electrical  conductivity  of  weak  solutions  of  acids  is 
due  almost  wholly  to  the  hydrion  they  contain,  and  measur- 
ing the  relative  strengths  of  acids  by  measuring  the  conduc- 
tivity of  the  solutions  has  practically  superseded  all  other 
methods,  especially  for  the  weaker  organic  acids. 

(Walker,  Introduction  to  Physical  Chemistry,  p.  313.) 

PROCESS   OF   NEUTRALIZATION. 

It  is  shown  by  conductivity  measurements  of  nearly  pure 
water  that  hydrogen  ions  and  hydroxyl  ions  can  only  exist 
beside  each  other  in  mere  traces,  if  at  all.  Hence  when  we 
bring  together  in  a  water-solution  two  electrolytes  (an  acid 
and  a  base)  which  yield  hydrogen  ions  and  hydroxyl  ions 
these  unite  to  form  water  according  to  the  equation 

H  +  OH  =  H,O. 

This  process  is  called  neutralization. 
(Nernst,  Theoretical  Chemistry,  pp.  517,  518.) 


96  LAWS  OF  PHYSICAL  SCIENCE 

NEUTRALIZATION  OF  AN  ACID  AND  A  BASE. 

When  a  strong  acid  is  mixed  with  a  strong  base  the 
hydrogen  ions  and  the  hydroxyl  ions  unite  almost  completely 
to  form  molecules  of  water  and  the  general  result  follows, 
that :  ' '  the  neutralization  of  a  strong  acid  by  a  strong  base 
must  always  exhibit  the  same  heat  of  reaction  " — about 
13,700  gram-calories. 

(See  Nernst,  Theoretical  Chemistry,  pp.  578-610.  Also 
Washburn,  Principles  of  Physical  Chemistry,  p.  243.  Also 
see  for  values  of  heats  of  neutralization,  Smithsonian 
Physical  Tables,  p.  213.) 

RELATIVE    AVIDITY    OF   ACIDS. 

Two  weak  acids  share  between  themselves  a  base  for 
which  both  are  competing  in  a  certain  definite  ratio.  The 
ratio  of  the  avidities  of  the  two  acids  is  (under  specified 
conditions)  equal  to  the  square  root  of  the  ratio  of  their 
dissociation-constants.  Thus, 

..  x     =  -*  /TT»      where  x  is  the  amount  of 


one  of  the  two  acids  neutralized  by  the  base  and  k  and  k' 
are  the  dissociation-constants  of  the  two  acids. 

(Walker,  Introduction  to  Physical  Chemistry,  pp.  314- 
316.  Also  Nernst,  Theoretical  Chemistry,  pp.  521-523.) 

CHEMICAL    COMBINATION    OF   THE   ELEMENTS. 

"  Combination  always  takes  place  between  certain  defi- 
nite and  constant  proportions  of  the  elements  or  between 
multiples  of  these. " 

The  composition  of  a  chemical  compound  can  be  learned 
by  (1)  bringing  the  component  elements  into  combination 
under  favorable  conditions:  synthesis,  or  (2)  by  separating 
it  into  its  component  elements :  analysis. 

(Roscoe  and  Schorlemmer,  Treatise  on  Chemistry,  Vol. 
I,  p.  72.  Also  Nernst,  Theoretical  Chemistry,  pp.  30,  31.) 


HEAT  AND  PHYSICAL  CHEMISTRY  97 

PERIODIC    SYSTEM    OF    CHEMICAL    ELEMENTS. 

The  properties  of  the  chemical  elements  are  periodic 
functions  of  their  atomic  weights. 

This  is  shown  by  means  of  a  systematic  arrangement  of 
the  elements  in  a  table  known  as  "Mendele Jeff's  Periodic 
System  of  the  elements. ' ' 

(See  table,  Nernst,  Theoretical  Chemistry,  p.  180.  Also 
see  table  published  July,  1915,  in  General  Electric  Review, 
by  the  Research  Laboratory  of  the  General  Electric 
Company.) 

LAW  OF  GAY-LUSSAC  AND  HUMBOLDT  ON  COMBINATIONS  OF 
GASES  BY  VOLUME. 

The  volumes  in  which  gaseous  substances  combine  chem- 
ically bear  a  simple  relation  to  one  another  and  to  the 
volume  of  the  resulting  product. 

(Roscoe  and  Schorlemmer,  Treatise  on  Chemistry,  Vol. 
I,  p.  75.) 

RICHTER'S  LAW. 

When  any  two  neutral  salts  undergo  double  decomposi- 
tion by  interchange  of  their  acid  and  basic  constituents  the 
two  new  salts  resulting  from  such  interaction  are  also  neutral 
in  character. 

(Roscoe  and  Schorlemmer,  Treatise  on  Chemistry,  Vol. 
I,  p.  33.) 

"  MOLECULAR  ROTATION  »  OF  AN  OPTICALLY  ACTIVE  LIQUID. 

A  plane-polarized  ray  in  passing  through  a  liquid 
optically  active  is  rotated  proportionally  both  to  the  length 
of  the  path  of  the  ray  through  the  liquid  and  to  the  density 
of  the  liquid.  Hence  when  the  actual  observed  rotation  is 
divided  by  the  density  and  the  length  of  the  path  the  specific 
rotary  power  of  the  liquid  is  obtained,  and  when  this  is 
multiplied  by  the  molecular  weight  the  molecular  rotation 
is  expressed. 

(Walker,  Introduction  to  Physical  Chemistry,  p.  158.) 


98  LAWS  OF  PHYSICAL  SCIENCE 

OUDEMAN'S   LAW  OF  OPTICAL   ROTATION. 

The  molecular  rotation  of  plane-polarized  light  by  salts 
of  an  optically  active  acid  or  base  always  tends  to  a  definite 
limiting  value  as  the  concentration  of  the  solution  dimin- 
ishes. This  regularity  is  known  as  Oudeman's  law. 

(Walker,  Introduction  to  Physical  Chemistry,  p.  174. 
Also  Nernst,  Theoretical  Chemistry,  p.  391.) 

RAOULT'S  LAW  FOR  DEPRESSION  OF  FREEZING-POINT. 

If  the  same  number  of  molecules  of  different  substances 
be  dissolved  in  a  given  number  of  molecules  of  the  solvent 
the  depressions  of  the  freezing-points  of  the  solutions  are 
equal. 

(New  Century  Dictionary  under  word  LAW.  Also 
Walker,  Introduction  to  Physical  Chemistry,  p.  210  et  seq. 
Also  Nernst,  Theoretical  Chemistry,  p.  146.) 

INFLUENCE  OF  PRESSURE  ON  THE  MELTING-POINT. 

The  melting-point  of  a  substance  (as  ice)  which  con- 
tracts on  liquefaction,  is  lowered  by  increase  of  pressure 
and  if  the  substance  expands  during  liquefaction  the 
melting-point  is  raised  by  increase  in  pressure. 

(Preston,  Theory  of  Heat,  pp.  341,  342.  Also  Nernst, 
Theoretical  Chemistry,  p.  67  et  seq.) 

DISSOLVED  SALTS   RAISE   BOILING-POINT. 

The  boiling-point  of  a  liquid  is  raised  by  dissolving  in 
it  a  salt.  Thus,  a  saturated  solution  of  common  salt  in  water 
boils  at  about  102°  C.  and  water  saturated  with  calcium 
chloride  boils  at  about  179°  C. 

The  boiling-point  is  lowered  when  a  gas  is  dissolved  in 
the  liquid. 

(Walker,  Introduction  to  Physical  Chemistry,  pp.  195, 
196.  Also  Ganot's  Physics,  art.  368.) 


HEAT  AND  PHYSICAL  CHEMISTRY  09 

FREEZING-POINT   OF  A   SOLUTION    (BLAGDEN'S   LAW). 

The  freezing-point  of  a  substance  is  lowered  by  dissolv- 
ing in  it  some  foreign  matter.  The  change  is  usually 
proportional  to  the  amount  of  dissolved-substance  in  the 
solvent,  although  sometimes  the  change  is  abnormally  great. 
This  is  the  case  with  substances  which  have  an  abnormal 
osmotic  pressure. 

(Consult  Walker,  Introduction  to  Physical  Chemistry, 
pp.  65,  195  and  406.) 

FUSION:    OF    CRYSTALLINE    SOLIDS    AND    PURE    METALS. 

1.  For  a  given  pressure  the  temperature  of  fusion  is  fixed, 

and  is  the  same  as  that  of  solidification. 

2.  While  fusion  or  solidification  is  taking  place  the  tempera- 

ture of  the  whole  mass  remains  constant. 

3.  During  fusion  heat  is  absorbed  by  the  substance  and  an 

equal  quantity  of  heat  is  disengaged  during  solidifica- 
tion. 
(Preston,  Theory  of  Heat,  p.  336.) 

THE  CRYOHYDRIC  TEMPERATURE. 

The  vryohydric  temperature  is  the  lowest  temperature 
that  can  be  produced  by  mixing  salt  and  ice  together.  No 
solution  of  salt  in  water  can  exist  in  a  stable  state  below  this 
temperature. 

(For  full  and  accurate  information  on  this  matter,  con- 
sult Walker,  Introduction  to  Physical  Chemistry,  Chap. 
VIII.) 

GIBBS'  CRITERIA  OF  THERMAL  EQUILIBRIUM. 

For  the  equilibrium  of  any  isolated  system  it  is  neces- 
sary and  sufficient  that,  if  the  total  energy  of  the  system  be 
constant,  the  variation  of  its  entropy  shall  either  vanish  or 
be  negative;  or,  if  the  total  entropy  of  the  system  be  con- 
stant, the  variation  of  its  energy  shall  either  vanish  or  be 
positive. 

(Gibbs,  ''On  the  Equilibrium  of  Heterogeneous  Sub- 
stances, "  Trans.  Connecticut  Acad.,  Vol.  3,  p.  109.) 


100  LAWS  OF  PHYSICAL  SCIENCE 

THE  PHASE-RULE. 

The  Phase-Rule,  due  to  Gibbs,  generalizes  the  condi- 
tions which  determine  the  equilibrium  of  any  system.  Let 
K  represent  the  number  of  components  (salt,  water,  etc.) 
and  let  i  be  the  total  number  of  phases  in  which  these  com- 
ponents are  present  (crystalline,  liquid,  vapor,  etc.).  The 
number  of  degrees  of  freedom,  or  independent  ways  in 
which  the  system  can  be  changed,  is  given  by 
n  =  K  +  2  -  i. 

(New  Century  Dictionary  under  words  PHASE  RULE. 
Also  Jones,  Elements  of  Physical  Chemistry,  p.  489  et 
seq.  Also  Walker,  Introduction  to  Physical  Chemistry, 
Chap.  XI,  pp.  103-117.  Also  Gulliver,  Metallic  Alloys,  pp. 
157-164.) 

LAW   OF   THE   MUTUALITY  OF  PHASES. 

"If  two  phases,  respecting  a  certain  definite  reaction, 
at  a  certain  temperature,  are  in  equilibrium  with  a  third 
phase,  then  at  the  same  temperature  and  respecting  the 
same  reaction,  they  are  in  equilibrium  with  each  other. ' ' 

(Nernst,  Theoretical  Chemistry,  p.  672.  See  also  pp. 
132,  137,  495  for  applications  of  law.) 

ACCELERATION  OF  CHEMICAL  REACTIONS  WITH  ELEVATION 
OF  TEMPERATURE. 

It  is  a  general  principle  of  chemical  kinetics  that  the 
velocity  with  which  a  chemical  system  proceeds  toward  its 
state  of  equilibrium  increases  very  greatly  with  increase  in 
temperature. 

(Nernst,  Theoretical  Chemistry,  pp.  679,  680.  Also 
Walker,  Introduction  to  Physical  Chemistry,  p.  300  et 
seq.) 


HEAT  AND  PHYSICAL  CHEMISTRY  101 

HEATS  OF  REACTION:  LAW  OF  CONSTANT  HEAT-SUMMATION. 

In  processes  which  occur  in  nature  the  associated  energy 
changes  may  be  discriminated  as : 

1.  Production  or  absorption  of  heat. 

2.  Performance  of  external  work. 

3.  Variation  of  the  internal  energy  of  a  system. 

In  a  chemical  system  the  sum  of  the  heat  produced  in  a 
reaction  and  the  external  work  performed  is  called  the ' '  heat 
of  reaction. ' '  This  heat  of  reaction  (also  heat  of  formation) 
may  be  either  positive  or  negative.  It  represents  the  change 
in  the  total  energy  of  the  chemical  system. 

The  total  heat  generated  in  a  chemical  reaction  is  entirely 
independent  of  the  steps  followed  in  passing  from  initial  to 
final  state  of  the  system,  and  this  principle — "the  law  of 
constant  heat-summation " — makes  it  possible  to  calculate 
heats  of  formation  for  steps  which  are  chemically 
impracticable. 

(Nernst,  Theoretical  Chemistry,  pp.  592,  597  et  seq. 
Also  Smithsonian  Physical  Tables,  p.  212.) 

HEAT  OF  FORMATION. 

By  "heat  of  formation  of  a  chemical  compound  is  meant 
the  quantity  of  heat  which  is  given  off  in  the  formation  of 
the  compound  from  its  component  elements. "  The  sum  of 
the  heats  of  formation  of  the  substances  formed  by  chemical 
reaction  minus  the  sum  of  the  heats  of  formation  of  the 
substances  used  up,  is  equal  to  the  heat  of  reaction. 

(Nernst,  Theoretical  Chemistry,  p.  606.  See  also  607, 
608,  where  it  is  shown  how  heats  of  formation  may  be  ob- 
tained from  heats  of  combustion.} 


102  LAWS  OF  PHYSICAL  SCIENCE 

CATALYSIS. 

Many  chemical  reactions  are  observed  to  take  place  at 
an  accelerated  rate  when  they  occur  in  the  presence  of 
certain  substances  which  themselves  suffer  no  chemical 
change.  Berzelius  gave  to  this  phenomenon  the  name  catal- 
ysis. The  name  means,  * '  an  increase  in  velocity  of  reaction 
caused  by  the  presence  of  substances  which  do  not  take  part 
in  it  (or  only  to  a  secondary  extent)  although  the  reaction 
is  capable  of  taking  place  without  their  presence/' 

A  substance  which  produces  catalysis  is  called  a  cata- 
lyser. 

(Nernst,  Theoretical  Chemistry,  p.  581.  Also  Wash- 
bum,  Principles  of  Physical  Chemistry,  pp.  274,  275.) 

CRYSTALLOIDS   AND   COLLOIDS. 

Investigations  of  the  phenomena  of  diffusion  show  that 
substances  can  be  divided  into  two  classes,  "  crystalloids " 
and  ' '  colloids. ' '  The  former  diffuse  more  rapidly  and  as  a 
rule  are  obtained  in  crystalline  form  while  the  latter  are 
amorphous.  Graham  named  the  process  for  separating  the 
two  classes  by  means  of  an  animal  membrane,  "dialysis." 

(Consult  Walker,  Introduction  to  Physical  Chemistry, 
p.  233.  Also  Ganot's  Physics,  art.  140.) 

ABSORPTION  OF  RADIANT  HEAT. 

It  is  a  general  principle  that  bodies  absorb  radiant  heat 
which  proceeds  from  heated  bodies  of  the  same  kind.  Also, 
"any  substance  is  particularly  transparent  to  radiation 
which  has  already  been  sifted  by  a  plate  of  that  substance. ' ' 

(Preston,  Theory  of  Heat,  p.  549.) 


HEAT  AND  PHYSICAL  CHEMISTRY  103 

ABSOLUTE    EMISSIVE    POWER    (Definition). 

The  absolute  emissive  power  of  a  body  for  a  particular 
wave-length  is  the  energy  at  that  wave-length  radiated  per 
second  by  unit  surface  at  temperature  1°  absolute  to  sur- 
rounding enclosure  at  absolute  zero. 

(Preston,  Theory  of  Heat,  p.  588.) 

MONOCHROMATIC   EMISSIVE  POWER. 

The  monochromatic  emissive  power  of  a  body  is  the  ratio 
of  the  energy,  having  wave-lengths  lying  between  A  and  A. 
H-  dA,  which  it  radiates  at  absolute  temperature  T  to  the 
energy  of  the  same  wave-lengths  which  a  perfectly  black 
body  radiates  at  the  same  temperature  and  under  exactly 
similar  circumstances. 

Call  JA'  the  energy  of  these  wave-lengths  radiated  by  a 
unit  surface  of  the  body  in  the  unit  of  time  and  call  JA  the 
energy  radiated  by  a  black-body  under  the  same  circum- 

Ja 

stances.    Then   y-  =  e^,  the  monochromatic  emissive  power 

JA 

of  the  body.  It  is  in  general  a  function  of  the  absolute 
temperature  T,  the  wave-length  A  and  varies  with  the  nature 
of  the  body. 

(Consult  Preston,  Theory  of  Heat,  p.  587.  Also  Chwol- 
son,  Traite  de  Physique,  Vol.  II,  Part  8,  pp.  55,  56.  For 
experimental  methods  and  values  of  emissivity,  see  Bulletin 
of  the  Bureau  of  Standards,  Vol.  II,  p.  41,  p.  591  and  p.  607 ; 
articles  by  Burgess,  Waltenberg  and  Foote.) 


104  LAWS  OF  PHYSICAL  SCIENCE 

ABSORPTIVE   POWER  AND  DEFINITION   OF  A  BLACK-BODY. 

Let  radiation  of  a  given  wave-length  A  and  of  energy  E 
per  unit  volume  fall  upon  a  body.  Denote  this  radiant 
energy  by  E  ^ .  Then  if  a  portion  E  '  of  this  radiant 
energy  is  absorbed  by  the  body,  the  absorptive  power  of  the 
body  for  wave-length  A  is, 


If  this  fraction  is  unity  for  all  values  of  A  the  body  is 
called,  according  to  Kirchhoff,  a  perfectly  "black-body/7 

No  body  has  been  found  which  is  perfectly  black  as 
Kirchhoff  defines  black.  If,  however,  any  body  is  placed 
in  an  enclosure  the  walls  of  which  are  at  a  uniform  tempera- 
ture, the  body  will  finally  assume  the  temperature  of  the 
enclosure  and  when  it  does  it  will  emit  in  quantity  and 
quality  (i.e.,  wave-length)  as  much  radiation  as  it  receives. 
It  will  then  be  indistinguishable  to  the  eye  from  the  neigh- 
boring bodies  and  is  said  to  be  at  "black-body"  temperature. 

(Consult  Burgess  and  LeChatelier,  Measurement  of 
High  Temperatures,  Chap.  VI.) 

KIRCHHOFF'S   LAW  REGARDING  RADIATION. 

For  radiations  of  the  same  wave-length  and  the  same 
temperature,  the  ratio  of  the  emissive  and  absorptive  powers 
is  the  same  for  all  bodies  and  is  equal  to  the  emissive  power 
of  a  perfectly  black-body.  In  symbols, 

c^  ^^L, 

where  ex  is  the  emissive  power  of  the  body  for  wave-length 
A,  and  ax  is  its  absorptive  power  for  wave-length  A,  andCx 
is  the  emissive  power  of  a  black-body. 

(Preston,  Theory  of  Heat,  p.  588.     Also  Burgess  and 


HEAT  AND  PHYSICAL  CHEMISTRY  105 

LeChatelier,  Measurement  of  High  Temperatures,  pp.  243- 
245.  Also  Chwolson,  Trait  e  de  Physique,  Vol.  II,  Part  8, 
pp.  57-59.) 

PROPOSITIONS  DEDUCIBLE  FROM  KIRCHHOFF'S  RADIATION  LAW. 

1.  The  emissive  power  of  an  absolutely  black-body  is  the 

greatest  emissive  power  possible. 

2.  Every  body  absorbs  the  rays  which  it  emits  at  a  given 

temperature — but  every  body  does  not  necessarily  emit 
all  rays  which  it  absorbs  at  a  given  temperature. 

3.  Every  body  can  absorb  rays  which  it  emits  at  a  given 

temperature,  and  it  can  also  absorb  other  rays,  pro- 
vided that  among  these  latter  there  are  rays  which  a 
black-body  emits  at  the  given  temperature. 

4.  Every  body  which  emits,  at  a  given  temperature  and 

under  given  conditions  in  a  particular  direction  (under 
a  given  angle  with  the  normal),  rays  of  wave-length  A 
and  of  a  definite  type  of  vibration  (character  of 
polarization),  absorbs,  at  the  same  temperature  and 
under  the  same  conditions,  rays  of  the  same  wave- 
length and  the  same  type  of  vibration  which  fall  upon 
it  in  the  same  direction. 

5.  The  ratio  of  the  emissive  to  the  absorptive  power,  which 

is  the  same  for  all  bodies  for  the  same  given  values  of 
temperature  and  wave-length,  does  not  depend  upon 
the  kind  of  vibration;  namely,  upon  the  character  of 
the  rays  emitted  and  absorbed,  in  respect  to  their 
polarization. 

6.  The  law  of  Kirchhoff  applies  to  any  composite  flux  of 

calorific  energy  (where  the  wave-lengths  are  comprised 
between  any  arbitrary  limits  A±  and  A2)  if  the  integral 
absorption  be  referred  to  a  flux,  which  has  for  its 
source  an  absolutely  black  body  taken  at  the  same 
temperature  as  the  bodies  which  are  to  be  intercom- 
pared. 


106  LAWS  OF  PHYSICAL  SCIENCE 

7.  Kirchhoff  's  law  holds  true  for  a  composite  flux  when  two 

given  bodies  are  at  the  same  temperature  and  when 
each  of  them  acts  as  source  of  the  flux,  the  integral 
absorption  of  which,  by  the  other  body,  is  measured. 

8.  In  a  closed  space  all  parts  of  which  are  at  the  same 

temperature  all  bodies  inside  and  the  walls  of  the 
enclosure  itself  produce  definite  radiation  which  is 
identical  with  the  radiation  from  an  absolutely  black 
body. 

(For  full  explanation  and  proof  of  the  above  proposi- 
tions, consult  Chwolson,  Traite  de  Physique,  Vol.  II,  Part 
8,  pp.  59-70.) 

INTENSITY  OF  RADIANT  HEAT. 

The  quantity  of  heat  proceeding  from  a  point-source  of 
heat  which  is  received  on  a  unit  surface  in  the  unit  of  time 
may  be  called  the  intensity  of  radiant  heat.  It  varies  with 
the  temperature  of  the  source  and  is  inversely  proportional 
to  the  square  of  the  distance  from  the  source. 

(Ganot's  Physics,  art.  420.) 

HEAT-RADIATION  AT  AN  OBLIQUE  ANGLE. 

The  intensity  of  oblique  rays  of  radiant  heat  is  propor- 
tional to  the  cosine  of  the  angle  which  these  rays  form  with 
the  normal  to  the  surface.  This  ' '  law  of  the  cosine  ' '  is  not, 
however,  general. 

Radiant  heat  is  only  one  section  of  the  spectrum  which 
extends  from  wave-lengths  shorter  than  those  of  ultraviolet 
light  to  the  longest  waves  observed.  Hence,  the  laws  of 
reflection  and  refraction  are  the  same  for  rays  of  radiant 
heat  as  they  are  for  light. 

(Fourier,  The  Analytical  Theory  of  Heat,  p.  34.  Also 
Ganot's  Physics,  art.  420.) 


HEAT  AND  PHYSICAL  CHEMISTRY  107 

STEFAN-BOLTZMANN  RADIATION  LAW. 

The  energy  radiated  in  unit  time  by  a  black-body  is 
proportional  to  the  fourth  power  of  the  absolute  tempera- 
ture, or 

E=K  (TVT4). 

where  E  is  the  total  energy  radiated  by  the  body  at  absolute 
temperature  T  to  the  walls  of  an  enclosure  at  absolute 
temperature  T0  and  K  is  a  constant. 

(Preston,  Theory  of  Heat,  p.  590,  and  pp.  596-598.  Also 
Bulletin  of  the  Bureau  of  Standards,  Vol.  I,  p.  198.  Also 
Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p.  71  et 
seq.) 

PRESSURE   OF  RADIATION. 

When  radiant  energy  is  incident  perpendicularly  on  a 
plane-surface  which  is  absolutely  black,  it  exerts  a  pressure 
on  the  surface  equal  to  the  density  of  the  energy  of  radia- 
tion. If  the  body  is  perfectly  reflecting  the  pressure  is 
twice  as  great. 

(Maxwell,  Electricity  and  Magnetism,  Vol.  II,  art.  792. 
Also  Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p.  84.) 

WIEN'S  DISPLACEMENT  LAW   (ist  statement). 

"If  radiation  of  a  particular  wave-length  corresponding 
to  a  definite  temperature  is  adiabatically  altered  to  another 
wave-length,  then  the  temperature  changes  in  the  inverse 
ratio,"  Or, 

a  constant 
T= — 

(Consult  Preston,  Theory  of  Heat,  p.  600.  Also  p.  602 
for  confirmatory  experiments.) 


108  LAWS  OF  PHYSICAL  SCIENCE 

WIEN'S  DISPLACEMENT  LAW  (and  statement). 

"When  the  temperature  increases,  the  wave-length  of 
every  monochromatic  radiation  diminishes  in  such  a  way 
that  the  product  of  the  temperature  and  the  wave-length  is 
a  constant.'*  Or, 

AT  «  A0T0. 
Hence  for  the  wave-length  of  maximum  energy 

^T  =  a  constant. 

(Bulletin  of  the  Bureau  of  Standards,  Vol.  I,  No.  2,  p. 
202.) 

WIEN'S  LAW  OF  RADIATION  OF  WAVE-LENGTH  OF  MAXIMUM 
ENERGY. 

The  energy  radiated  from  a  black-body  source  which 
corresponds  to  the  wave-length  having  a  maximum  energy 
is  proportional  to  the  fifth  power  of  the  absolute  tempera- 
ture. Or, 

En,a,   =    BT% 

where  T  is  the  absolute  temperature  and  B  is  a  constant. 
(Bulletin  of  the  Bureau  of  Standards,  Vol.  I,  p.  202. 
Also  Preston,  Theory  of  Heat,  pp.  601,  610.     Also  Chwol- 
son,  Traite  de  Physique,  Vol.  II,  part  8,  p.  73.) 

WIEN'S  LAW  OF  SPECTRAL  DISTRIBUTION  OF  ENERGY, 

C2 

J  =  Ci  X~5e       XT 

where  J  =  the  energy  corresponding  to  wave-length  A, 

T  =  absolute  temperature  of  the  radiating  black-body, 
e  =  base  of  the  natural  system  of  logarithms  and 
GI  and  c2  are  constants. 
(Bulletin  of  the  Bureau  of  Standards,  Vol.  I,  p.  204. 

For  use  of  the  equation  in  pyrometry,  see  p,  210.) 


HEAT  AND  PHYSICAL  CHEMISTRY  109 

PLANCK'S  LAW  OF  SPECTRAL  DISTRIBUTION  OF  ENERGY. 


where  J  is  the  energy  corresponding  to  wave-length  X,  T  is 
the  absolute  temperature,  e  the  base  of  the  natural  system 
of  logarithms  and  c±  and  c2  are  constants. 

Planck's  law  agrees  with  experiment  better  than  Wien's 
and  holds  well  for  a  wide  variation  in  A. 

(Bulletin  of  the  Bureau  of  Standards,  Vol.  I,  p.  206.) 

PREVOST'S  THEORY  OF  EXCHANGES. 

If  two  bodies  are  associated  in  such  a  manner  that  one 
receives  the  radiation  of  the  other,  each  radiates  indepen- 
dently and  the  temperature  of  either  will  fall  if  it  radiates 
more  energy  than  it  absorbs. 

(Ganot's  Physics,  art.  421.) 

THE  QUANTUM  HYPOTHESIS. 

The  only  hypothesis  which  has  satisfactorily  accounted 
for  the  laws  of  radiation  from  black  bodies  and  many 
phenomena,  such  as  the  emission  of  electrons  from  illumi- 
nated metals,  has  been  based  on  the  supposition  that  radiant 
energy  is  absorbed  or  emitted  or  both  not  continuously,  but 
by  discontinuous,  discrete  units  of  magnitude  proportional 
to  the  frequency  of  the  radiation. 

(Planck,  Vorlesungen  uber  die  Theorie  der  Warme- 
strahlung.) 

LAW    OF  PHOTO-CHEMICAL    REACTION. 

Abundant  research  has  led  to  the  result  that  when  a 
photo-chemical  system  is  illuminated  the  resultant  action 
depends  both  on  the  light-intensity  and  the  time  the  illumi- 
nation continues.  Hence  the  law:  "When  light  of  the  same 
kind  is  used,  the  photo-chemical  action  depends  solely  on  the 
product  of  the  intensity  and  the  duration  of  the  exposure." 
This  law  applies  in  photography  and  for  X-Ray  exposures. 

(Nernst,  Theoretical  Chemistry,  p.  786.) 


110  LAWS  OF  PHYSICAL  SCIENCE 

CRYSTALS:  THE  LAW  OF  INTERFACIAL  ANGLES. 

A  crystal  is  a  homogeneous  body  the  various  physical 
properties  of  which  are  differently  manifested  when  con- 
sidered along  different  lines  radiating  from  any  point  in 
the  body. 

The  fundamental  law  of  crystallography  states  that, 
*  *  the  inclination  of  two  definite  crystal  planes  to  each  other, 
for  the  same  substance,  and  measured  at  the  same  tempera- 
ture, is  constant  and  independent  of  the  size  and  develop- 
ment of  the  planes. ' '  This  law  is  not,  however,  quite  rigid. 

(Nernst,  Theoretical  Chemistry,  pp.  72,  73.  Also 
Spencer,  The  World's  Minerals,  p.  15.) 

NEUMANN'S  LAW  OF  CRYSTAL-ZONES. 

A  set  of  planes  of  a  crystal  which  intersect  in  such 
manner  that  the  lines  of  intersection  are  parallel  to  one 
another  is  called  a  zone  and  this  common  direction  is  called 
the  " zonal  axis."  Otherwise  stated  a  crystal  has  a  girdle 
of  faces  (a  zone)  the  edges  of  which  form  parallel  lines. 

It  is  a  fundamental  law  of  crystallography  that  all  planes 
which  can  occur  on  a  crystal  are  related  to  each  other  in 
zones  or  from  any  four  planes,  no  three  of  which  lie  in  any 
one  zone,  all  crystal  planes  can  be  derived  by  means  of 
zones. 

(Nernst,  Theoretical  Chemistry,  p.  73.  Also  Spencer, 
The  World's  Minerals,  pp,  17,  18.) 


V 
ELECTRICITY  AND  MAGNETISM 


ELECTRICITY  AND  MAGNETISM 

ELECTRIFICATION  PRODUCED  BY  FRICTION. 

If  two  unlike  substances  which  are  insulators  are  rubbed 
together  and  then  separated  they  are  found  to  be  electrified 
with  equal  quantities  of  electricity,  the  one  substance  with 
positive,  or  vitreous,  and  the  other  with  negative,  or  resinous, 
electricity. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  27.) 

ELECTRIFICATION  PRODUCED   BY  INDUCTION. 

When  an  electrified  body  is  suspended  in  a  hollow  con- 
ducting vessel  without  touching  it,  the  outside  of  the  vessel 
acquires  electrification  of  the  same  sign  as  the  electrified 
body,  and  the  inside  of  the  vessel  acquires  electrification  of 
the  opposite  sign.  The  electrification  of  the  vessel  thus 
produced  is  called  electrification  by  induction. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  28.  Also  Ganot's  Physics,  art.  764.) 

ELECTRIFICATION   BY   CONDUCTION. 

When  an  originally  unelectrified  and  insulated  metal 
body  is  connected  with  an  electrified  body  by  means  of  a 
conducting  wire,  which  is  itself  insulated,  the  first  body 
becomes  electrified  by  a  passage  of  electricity  over  the  wire. 
This  passage  is  called  electrification  by  conduction. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  29,) 


113 


114  LAWS  OF  PHYSICAL  SCIENCE 

CONDUCTORS  AND  INSULATORS. 

At  low  and  at  ordinary  temperatures  materials  are 
separable  into  two  fairly  distinct  classes :  those  which  readily 
conduct  electricity,  called  conductors,  and  those  which  con- 
duct electricity  very  slightly,  or  not  at  all,  called  insulators. 
Above  1200°  or  1500°  C.  the  distinction  between  the  two 
classes  disappears  rapidly  as  the  temperature  increases. 

(See  "  Methods,  Data  and  New  Apparatus  for  Measuring 
Electrical  Conductivity  above  1500°  C.  of  Vapors  at  Normal 
Pressure,"  by  E.  F.  Northrup,  Jour,  of  the  Franklin  Insti- 
tute, March,  1915.  Also  Ganot's  Physics,  art.  743.) 

EQUALITY  OF  POSITIVE  AND  NEGATIVE  ELECTRICITY. 

When  electrification  is  excited  by  any  means  the  quan- 
tities of  positive  and  negative  electricity  produced,  or  re- 
vealed, are  always  equal. 

Modern  theory  asserts  that  the  ultimate  unit  of  negative 
electricity  is  a  definite  quantity  of  electricity  called  an  elec- 
tron. Millikan  concludes  from  experiments  that  the  charge 
carried  by  an  electron  is  4.774  X  10~10  electrostatic  unit. 
The  ratio,  £ ,  of  the  charge  (electrostatic  units)  to  the 

m 

mass  (grams)  of  an  electron  is  5.31  X  1017,  provided  the 
velocity  of  the  electron  is  a  small  fraction  of  the  velocity 
of  light. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
10,  See  also  Campbell,  Modern  Electrical  Theory,  pp. 
'25-28,  78-80.  Also  Smithsonian  Physical  Tables,  p.  342.) 

LAW    OF    REPULSION     OF    ELECTRIC     CHARGES. 

When  two  charged  bodies  are  at  a  distance  r  apart,  r 
being  very  large  compared  with  the  greatest  linear  dimen- 
sions of  either  of  the  bodies,  the  repulsion  between  them  is 
proportional  to  the  product  of  their  charges  and  inversely 
proportional  to  the  square  of  the  distance  between  them. 


ELECTRICITY  AND  MAGNETISM  115 

The  repulsion  between  two  charges  Q±  and  Q2  in  air  is, 
in  electrostatic  units,  F  =-^-|-  dynes. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
12.     Also,   for  simple  proof,   see  Lommel,   Experimental 
Physics,  p.  285.    Also  Ganot's  Physics,  art.  753.) 

POTENTIAL. 

The  work  which  must  be  done  by  an  external  agent  to 
bring  a  unit  of  positive  electricity,  by  any  path,  from  an 
infinite  distance  (or  from  a  place  where  the  potential  is 
zero)  to  a  given  point  in  space  is  called  the  potential  at 
the  point. 

A  body  charged  positively  always  tends  to  move  from 
places  of  greater  to  places  of  less  positive  potential,  and  a 
body  charged  negatively  always  tends  to  move  in  the  oppo- 
site direction. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  arts.  45,  70.) 

ELECTROMOTIVE    FORCE. 

The  difference  of  potential  between  two  conductors  or 
between  two  points  in  space  or  between  two  points  on  a 
body  is  equal  to  the  electromotive  force  (e.m.f.)  between 
them.  The  e.m.f.  between  the  two  points  in  a  circuit  is  equal 
to  the  product  of  the  current  and  the  ohmic  resistance 
between  the  two  points. 

When  a  charge  e  moves  along  a  given  path  from  a  point 
A  to  a  point  B  and  work  W  is  done  by  the  electric  force, 
such  that  W  =  Ee,  the  quantity  E  is  called  the  total  elec- 
tromotive force  acting  between  the  points  A  and  B.  In 
electrostatics  E  and  Vj-V,,  the  potential  difference  between 
the  points,  are  identical  quantities. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  arts.  45,  49,  69,  241.) 


116  LAWS  OF  PHYSICAL  SCIENCE 

POTENTIAL  DUE  TO  A  SYSTEM  OF  POINT-CHARGES. 

When  any  number  of  electrified  points  having  charges 
ei>  e2>  es>  etc.,  are  distributed  through  space,  then  if 
ri>  r2>  ra>  etc.,  are  the  distances  of  these  points  respectively 
from  a  point  P  in  space,  the  potential  at  P  due  to  the 
system  is, 


(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol.  I, 
art.  73.) 

FORCE  BETWEEN  CHARGED  BODIES  VARIED  BY  THE  MEDIUM. 

When  the  charges  are  given,  the  mechanical  forces  on 
bodies  in  an  electric  field  are  diminished  by  the  interposi- 
tion of  a  medium  with  a  larger  specific  inductive  capacity. 
Thus  the  force  between  two  point-charges  Qx  and  Q2,  a 
distance  r  apart  in  a  medium  of  specific  inductive  capacity 
K,is, 


(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
129.) 

ELECTRIC    EQUILIBRIUM. 

A  conductor  can  only  be  in  electric  equilibrium  when 
every  point  in  it  is  at  the  same  potential.  This  potential  is 
called  the  Potential  of  the  Conductor. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol.  I, 
art.  45.  Also  consult,  for  an  extended  and  clear  exposition 
of  the  facts  and  laws  of  electricity  in  equilibrium,  Chwolson, 
Traite  de  Physique,  Vol.  4,  Part  10,  Chap.  I.  See  p.  81.) 

ELECTRIC    ABSORPTION. 

The  phenomenon  of  electric  absorption  is  not  an  actual 
absorption  of  electricity,  for  if  a  condenser  is  in  the  interior 


ELECTRICITY  AND  MAGNETISM  117 

of  a  hollow  electric  conductor  there  is  no  alteration  in  its 
surface  electrification  by  the  "absorption"  taking  place  in 
the  condenser.  Or,  as  stated  by  Faraday :  '  'It  is  impossible 
to  charge  matter  with  an  absolute  and  independent  charge 
of  one  kind  of  electricity. ' ' 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  arts.  53,  54.) 

ELECTRIC  INTENSITY  INSIDE  AN  INCLOSED  CONDUCTING  SURFACE. 

There  is  no  electric  intensity  or  lines  of  electric  force  on 
the  inside  of  a  surface  which  is  conducting  and  which  com- 
pletely incloses  a  portion  of  space,  however  highly  this 
surface  is  charged,  if  there  is  no  charged  body  in  the  space. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
21,  22,  29.  Also  Ganot's  Physics,  art.  765.) 

GAUSS'    THEOREM.  y, 

"The  total  normal  electric  induction  over  any  closed    ^ 
surface  drawn  in  the  electric  field  is  equal  to/47r  times)  the 
total  charge  of  electricity  inside  the  closed  surface." 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
14.) 

COULOMB'S    LAW. 

The  electric  intensity  of  a  point  p  close  to  the  surface 
of  a  conductor  surrounded  by  air  is  at  right  angles  to  the 
surface,  It  is  equal  to  4?r  a  where  o-  is  the  surface  density  of 
the  electrification.  If  the  surface  of  the  conductor  is  in 
contact  with  a  dielectric  of  specific  inductive  capacity  K, 
then  the  electric  intensity  at  the  point  p  is, 


(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 

36,  122.) 


118  LAWS  OF  PHYSICAL  SCIENCE 

ENERGY  OF  A  SYSTEM  OF  CONDUCTORS. 

The  potential  energy  of  a  system  of  charged  electric 
conductors  placed  in  an  electric  field  which  arises  from  the 
charges  on  the  conductors  of  the  system  is  equal  to  one-half 
the  sum  of  the  products  obtained  by  multiplying  the  charge 
on  each  conductor  by  its  potential,  or, 

E  =  K^QV, 
where  Q  is  a  charge  and  V  the  potential  of  that  charge. 

(Thomson,  Elements  of  Electricity  and  Magnetism, 
p.  37.) 

ELECTRICITY  AND  AN  INCOMPRESSIBLE  FLUID  COMPARED. 

"The  motions  of  electricity  are  like  those  of  an  incom- 
pressible fluid,  so  that  the  total  quantity  within  an  imaginary 
fixed  closed-surface  remains  always  the  same." 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  61.) 

WORK   DONE   IN  A   DISPLACEMENT  OF  AN  ELECTRIFIED  SYSTEM. 

The  work  done  by  the  electric  forces  during  the  displace- 
ment of  an  electrified  system,  when  the  potentials  are  main- 
tained constant,  is  equal  to  the  increment  of  the  electric 
energy.  The  work  done,  therefore,  by  a  battery  which 
maintains  the  potentials  constant  is  twice  the  work  done  by 
the  system  during  its  displacement. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  93c.) 

POTENTIAL    ENERGY-CHANGES. 

If  any  small  displacement  of  a  system  of  electrified  con- 
ductors takes  place  the  diminution  in  the  electric  energy  of 
the  system,  when  the  charges  are  kept  constant,  is  equal  to 
the  increase  in  the  potential  energy  when  the  same  displace- 
ment takes  place  and  the  potentials  are  kept  constant. 

(Thomson,  Elements  of  Electricity  and  Magnetism, 
p.  54.) 


ELECTRICITY  AND  MAGNETISM  119 

MECHANICAL  FORCE  AT  THE  SURFACE  OF  A  CHARGED  CONDUCTOR. 

The  surface  of  every  charged  conductor  is  subject  to  a 
mechanical  force  which  acts  outward  along  the  normal.  The 
value  of  the  force  per  unit  area  of  the  surface  of  the  con- 
ductor for  any  dielectric  surrounding  the  conductor,  is, 
F  =  y2  Ro-,  and,  if  the  dielectric  surrounding  the  conductor 
is  air, 

R2 

F  =  __,=27rff2. 

Here  K  is  the  electric  intensity  and  o-  is  the  density  of  the 
surface  electrification.    The  maximum  value  of  P  in  air  at 

1  CM 

normal  pressure  and  15°  C.  is  about    -^  dynes  per  square 
centimeter,  which  is  a  pressure  of  about  0.3  mm.  of  mercury. 
(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
58,  59.) 

PASCHEN'S  LAW  FOR  SPARKING  POTENTIALS  IN  A  GAS. 

The  sparking  potential  between  electrodes  in  a  gas  de- 
pends on  the  length  of  the  spark-gap  and  the  pressure  of 
the  gas  in  such  a  way  that  it  is  directly  proportional  to  the 
mass  of  gas  between  the  two  electrodes.  Or  we  can  consider 
the  sparking  potential  as  a  function  of  the  pressure  X  the 
density  of  the  gas. 

(Thomson,  Conduction  of  Electricity  Through  Gases, 
pp.  451-460.) 

STATE  OF  ELECTRIC  STRESS  IN  A  MEDIUM. 

When  bodies  in  a  dielectric  medium  are  electrified,  "at 
every  point  of  the  medium  there  is  a  state  of  stress  such 
that  there  is  tension  along  the  lines  of  force  and  pressure 
in  all  directions  at  right  angles  to  these  lines,  the  numerical 
magnitude  of  the  pressure  being  equal  to  that  of  the  tension, 
and  both  varying  as  the  square  of  the  resultant  force  at  the 
point. ' ' 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  109.) 


120  LAWS  OF  PHYSICAL  SCIENCE 

ELECTRIC    DISPLACEMENT. 

The  whole  quantity  of  electricity  Q,  which  is  displaced 
through  a  unit  area,  when  the  electric  intensity  E  is  normal 

T£ 

to  the  area,  is    Q  =  -j^-R;  where  K  is  the  specific  inductive 

capacity  of  the  dielectric. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  62.  Compare  with  statement  in  Thomson,  Elements 
of  Electricity  and  Magnetism,  p.  122.) 

ENERGY    OF    ELECTRIC    FIELD. 

The  electric  field  represents  a  certain  store  of  energy  in 
the  medium.  If  E  is  the  intensity  of  the  field  the  stored 

•rr 

energy  per  unit  volume  is  E  =  —  R2,    where  K  is  the  specific 

871" 

inductive  capacity  of  the  medium. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
70-72,  and  123.) 

SPECIFIC    INDUCTIVE    CAPACITY. 

The  charge  of  electricity  which  a  condenser  will  hold 
when  charged  to  a  given  potential  is  dependent  upon  the 
nature  of  the  dielectric.  The  ratio  of  the  charge  which  the 
condenser  holds  when  a  given  dielectric  is  used,  to  the 
charge  it  holds  when  air  (or  a  perfect  vacuum)  is  the 
dielectric,  is  called  the  ' l  specific  inductive  capacity ' '  of  the 
dielectric.  This  ratio  is  either  equal  to  or  greater  than 
unity. 

(See  Thomson,  Elements  of  Electricity  and  Magnetism, 
Chap.  IV.  Also  Ganot's  Physics,  art.  783.) 

CONDUCTORS  AND  DIELECTRICS  IN  AN  UNUNIFORM 
ELECTRIC  FIELD. 

A  conductor  placed  in  an  ununiform  electric  field  tends 
to  move  from  the  weak  to  the  strong  parts  of  the  field;  so 


ELECTRICITY  AND  MAGNETISM  121 

likewise  does  a  dielectric  surrounded  by  one  of  smaller 
specific  inductive  capacity. 

(Thomson,  Elements  of  Electricity  and  Magnetism, 
p.  128.) 

ELECTRIC   INTENSITY  INSIDE   AND   OUTSIDE    CONDUCTORS 
AND    DIELECTRICS. 

The  electric  intensity  inside  a  conductor  placed  in  an 
electric  field  vanishes,  and  just  inside  a  dielectric  of  greater 
specific  inductive  capacity  than  the  surrounding  medium  the 
electric  intensity  is  less  than  that  just  outside. 

(Thomson,  Elements  of  Electricity  and  Magnetism, 
p.  128.) 

EARNSHAW'S    THEOREM    ON    STABILITY. 

If  a  charged  body  is  placed  in  an  electric  field  and  is 
altogether  free  to  move,  it  is  always  in  unstable  equilibrium 
in  respect  to  translational  motion. 

(For  proof,  see  Maxwell,  Treatise  on  Electricity  and 
Magnetism,  Vol.  I,  art.  116.  Also  Jeans,  Electricity  and 
Magnetism,  p.  165.) 

CAPACITY  OF  CONDENSERS,  JOINED  IN  PARALLEL  AND  IN  SERIES. 

The  total  capacity  C  of  a  number  of  condensers  joined  in 
parallel  is  equal  to  the  sum  of  the  individual  capacities 
d,  C2,  C3,  etc.,  and  the  reciprocal  of  the  total  capacity  of  a 
number  of  condensers,  joined  in  series,  is  equal  to  the  sum 
of  the  reciprocals  of  the  individual  capacities.  Thus  for 
parallel  combination 

C  =  Ci  +  C2  +  C,  +  etc. 
and  for  series  combination 


(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
110-113.    Also  Ganot's  Physics,  art.  799.) 


122  LAWS  OF  PHYSICAL  SCIENCE 

MAGNETISM   (Definitions). 

1.  The  magnetic  moment  of  a  magnet  is,  M  =  ml  where  m 

is  the  strength  of  its  pole  and  1  the  distance  between  its 
poles. 

2.  The  intensity  of  magnetization  of  a  magnetizable  sub- 

stance, uniformly  magnetized,  is  I  =  -y-,  where  V  is 

its  volume  and  M  is  its  magnetic  moment. 
(Consult  Jeans,  Electricity  and  Magnetism,  pp.  355, 
358.) 

3.  A  magnetic  shell  is  a  thin  sheet  of  magnetizable  substance 

magnetized  at  each  point  in  the  direction  of  the  nor- 
mal to  the  sheet  at  that  point. 

4.  The  strength  of  a  magnetic  shell  is  the  intensity  I  of  its 

magnetization  times  t  its  thickness.    Thus, 

#  =  It  =  —  -=  magnetic  moment  per  unit  area. 

5.  The  magnetic  potential  at  a  point  due  to  a  magnetic  shell 

of  uniform   strength  is  V  =  0<a,  where  w  is  the  solid 
angle  at  the  point  subtended  by  the  contour  of  the  shell. 
(Consult  Jeans,  Electricity  and  Magnetism,  pp.   365, 
366.) 

FORCE  ACTING  BETWEEN  MAGNETIC  POLES. 

The  force  between  two  magnetic  poles  in  air  is  in  the 
straight  line  joining  them,  and  is  numerically  equal  to  the 
product  of  the  strengths  of  the  poles  divided  by  the  square 
of  the  distance  between  them. 


where  m  and  m'  are  the  strengths  of  the  poles  and  r  is  the 
distance  between  them.  If  m  and  m'  are  of  unlike  sign  the 
force  is  an  attraction,  if  of  like  sign  the  force  is  a  repulsion. 
The  law  as  here  stated  assumes  that  the  strength  of  each 
pole  is  measured  in  terms  of  a  unit,  the  magnitude  of  which 
is  deduced  from  the  terms  of  the  law. 


ELECTRICITY  AND  MAGNETISM  123 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  arts.  373,  374.  For  Gauss'  proof  of  law,  see  Thomson, 
Elements  of  Electricity  and  Magnetism,  p.  206.  Also 
Ganot's  Physics,  arts.  716-720.) 

t-i- 

POTENTIAL    DUE    TO    A    MAGNETIC    SOLENOID. 

A  "Magnetic  Solenoid"  is  a  filament  of  magnetic  matter 
so  magnetized  that  its  strength  is  the  same  at  every  trans- 
verse section.  The  Magnetic  Potential  due  to  a  magnetic 
solenoid  depends  only  on  its  strength  and  the  position  of  its 
ends,  called  its  poles, 


where  m  is  the  strength  of  its  poles  and  rt  and  r2  are 
distances  from  the  positive  and  negative  poles  respectively 
to  the  point  where  the  potential  is  V. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  407.) 

TOTAL  CHARGE  OF  MAGNETISM. 

The  total  charge  of  magnetism  reckoned  algebraically  on 
any  magnet  is  zero. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
190.  Also  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  II,  art.  377.  Also  Ganot's  Physics,  art.  697.) 


124  LAWS  OF  PHYSICAL  SCIENCE 

MAGNETIC    FORCE    DUE    TO    A    MAGNET. 

If  two  points  P  and  Q  be  taken  equidistant  from  the 
center  of  a  bar-magnet,  the  point  P  being  in  the  line  of  its 
axis  and  the  point  Q  in  the  equitorial  plane  at  right  angles 
to  the  axis  of  the  magnet,  then  the  magnetic  force  at  P  is 
twice  the  magnetic  force  at  Q.  Thus,  if  OP  is  the  distance  to 
P  and  OQ  is  the  distance  to  Q  from  the  center  of  the  magnet, 

2M  M 

H,-  —  ,-ndHo-— , 

where  M  is  the  magnetic  moment  of  the  magnet. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
195-197.) 

MAGNETIC  FIELDS  DUE  TO  A  MAGNETIZED  SPHERE  AND  A 
SMALL  MAGNET  COMPARED. 

A  "uniformly  magnetized  sphere  produces  the  same 
effect  outside  the  sphere  as  a  very  small  magnet  placed  at 
its  center,  the  axis  of  the  small  magnet  being  parallel  to 
the  direction  of  magnetization  of  the  sphere,  while  the 
moment  of  the  magnet  is  equal  to  the  intensity  of  magneti- 
zation multiplied  by  the  volume  of  the  sphere. ' ' 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
224.) 

MAGNETISM    INDUCED    BY    A    MAGNETIC    FIELD. 

All  substances  which  are  measurably  diamagnetic  or 
paramagnetic  when  placed  in  a  magnetic  field  become 
charged  with  magnetism,  and  the  diamagnetic  substances 
tend  to  move  toward  weaker  portions  of  the  field  and  the 
paramagnetic  substances  toward  stronger  portions  of  the 
field. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
242.  Also  Ganot's  Physics,  arts.  700,  701.) 


ELECTRICITY  AND  MAGNETISM  125 

MAGNETIC  INDUCTION. 

Magnetic  induction  is  a  vector  quantity  and  is  the  sum 
of  two  vectors,  the  magnetic  force  H  and  4rr  times  the 
magnetization  I. 

Thus,  B  =  H  +  47rl. 

Tubes  of  magnetic  induction  are  always  continuous  and 
closed  tubes. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  400.  Also  Thomson,  Elements  of  Electricity  and 
Magnetism,  pp.  244-247.  Also  Jeans,  Electricity  and 
Magnetism,  p.  373.) 

SOME   RELATIONS   OF   MAGNETIC    QUANTITIES. 

Let  H  =  magnetic  force, 

B  =  magnetic  induction, 

I  =  magnetization, 

fjL  =  magnetic  permeability  and 

k  =  magnetic  susceptibility,  then 


H=  B  -4*1  = 


(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
247.  Also  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  II,  art.  438.) 


126  LAWS  OF  PHYSICAL  SCIENCE 

MAGNETIC    HYSTERESIS. 

Magnetic  hysteresis  is  a  phenomenon  which  results  from 
the  transformation  of  magnetic  energy  into  heat  when  the 
magnetization  of  a  substance  is  changed.  It  is  a  species  of 
molecular  friction  and  is  always  exhibited  when  cyclical 
reversals  of  magnetic  flux  in  a  magnetizable  substance  are 
produced.  It  is  analogous  with  friction  in  mechanics. 

C.  P.  Steinmetz  gives  for  the  loss  of  energy  due  to 
hysteresis  in  ergs  per  cycle  per  cm3, 

W  =  ?Bi-6, 

where  B  is  the  maximum  magnetic  induction  per  cm2  and  y 
is  the  "coefficient  of  hysteresis."  Steinmetz  gives  "n  = 
0.0025  as  a  fair  average  value  for  selected  steel. 

(Steinmetz,  Alternating  Current  Phenomena,  Chap.  X. 
See  p.  116.) 

MAGNETIC    AND    ELECTRIC    ANALOGUES. 

There  is  a  complete  analogy  between  the  disturbance  in 
the  distribution  of  an  electric  field  produced  by  the  presence 
of  uncharged  dielectrics  and  the  disturbance  of  a  magnetic 
field  produced  by  paramagnetic  or  diamagnetic  bodies  in 
which  the  magnetism  is  entirely  induced.  In  this  analogy 
the  magnetic  force  H  is  equivalent  to  the  electric  intensity 
E  and  the  magnetic  permeability  /A  is  equivalent  to  the 
specific  inductive  capacity  K. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
258.) 

ELECTROMOTIVE-FORCE    SERIES. 

Two  unlike  metals  when  immersed  in  an  acid  or  salt 
solution  have  acting  between  them  an  electromotive  force. 
The  potential-difference  between  the  two  metals  when  im- 
mersed in  a  given  solution  varies  with  the  nature  of  the 
metals. 

Metals  may  be  arranged  in  an  electromotive-force  series 


ELECTRICITY  AND  MAGNETISM  127 

in  which  the  most  electropositive  metal  begins  and  the  most 
electronegative  metal  ends  the  series. 

(Ganot's  Physics,  art.  817.  See  also  Smithsonian  Phys- 
ical Tables,  p.  267.) 

GALVANIC   POLARIZATION. 

When  a  chemical  system,  originally  in  equilibrium,  is 
electrolysed  the  decomposition  caused  by  the  electrolysis 
produces  a  displacement  of  equilibrium,  and  it  follows 
necessarily  that  the  current  passed  through  the  system  has 
to  overcome  an  opposing  electromotive  force.  The  develop- 
ment of  this  opposing  electromotive  force  in  a  galvanic 
cell  is  termed  "polarization." 

(Nernst,  Theoretical  Chemistry,  p.  739.  Also  Walker, 
Introduction  to  Physical  Chemistry,  pp.  374,  375.) 

CONSERVATION  OF  ENERGY  IN  ELECTROLYSIS:  THOMSON'S 
THEOREM. 

1 '  The  electromotive  force  of  an  electrochemical  apparatus 
is  in  absolute  measure  equal  to  the  mechanical  equivalent  of 
the  chemical  action  on  one  electrochemical  equivalent  of 
the  substance." 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  arts.  262,  263.  For  a  scientific  treatment  of  the  principles 
of  electrochemistry  and  an  analysis  of  the  above  statement, 
consult  Nernst,  Theoretical  Chemistry,  Chap.  VII,  pp.  731 
et  seq.) 

ELECTROLYSIS    LITTLE    AFFECTED    BY    PRESSURE. 

If  the  products  of  an  electrolysis  are  gases  which  obey 
Boyle's  law,  the  product  of  their  pressure  and  volume  will 
be  constant  at  a  given  temperature  and  the  e.m.f.  required 
for  electrolysis  is  nearly  independent  of  the  pressure. 
Electrolysis  of  dilute  acids,  therefore,  cannot  be  checked  by 
confining  the  gases  of  decomposition  in  a  small  space. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  art. 
263.) 


128  LAWS  OF  PHYSICAL  SCIENCE 

FARADAY'S    FIRST   LAW    OF   ELECTROLYSIS. 

"  The  quantity  of  an  electrolyte  decomposed  by  the 
passage  of  a  current  of  electricity  is  directly  proportional 
to  the  quantity  of  electricity  which  passes  through  it. ' ' 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
282.  Also  Ganot's  Physics,  art.  939.) 

FARADAY'S    SECOND    LAW    OF    ELECTROLYSIS. 

"If  the  same  quantity  of  electricity  passes  through  dif- 
ferent electrolytes  the  weights  of  the  different  ions  deposited 
will  be  proportional  to  the  chemical  equivalents  of  the  ions. ' ' 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
282.  For  a  full  account  and  a  bibliography,  consult  Chwol- 
son,  Traite  de  Physique,  Vol.  IV,  Part  10,  Chap.  V.  See  p. 
617  for  statements  of  the  two  laws  of  electrolysis.) 

A   PRINCIPLE    OF    ELECTROLYTIC    DECOMPOSITION. 

An  electrolytic  decomposition  can  only  proceed  when 
the  loss  of  energy  in  the  battery  which  supplies  current  to 
an  electrolytic  cell  is  greater  than  the  gain  of  energy  in 
the  electrolyte  of  the  electrolytic  cell.  An  action  contrary 
to  the  above  would  violate  the  principle  of  the  conservation 
of  energy. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
301.) 

THE    "FARADAY." 

The  Faraday  is  the  name  now  commonly  used  to  denote 
the  "  quantity!  of  electricity  associated  with  a  chemical 
equivalent  in  any  electrochemical  change. ' ' 

The  deposition  of  silver  from  a  solution  of  silver- 
nitrate  has  been  most  extensively  investigated.  G.  W.  Vinal 
and  S.  J.  Bates  writing  in  the  Bulletin  of  the  Bureau  of 
Standards,  Jan.  2,  1914,  give  as  the  value  of  the  Faraday 
for  silver, 


ELECTRICITY  AND  MAGNETISM  1» 

OOQUISOO      =  96494  intemational  Coulombs. 
(Nernst,  Theoretical  Chemistry,  pp.  727,  728.) 

OHM'S    LAW. 

The  current  which  flows  in  a  metallic  conductor  or  in  an 
electrolyte  is  proportional  to  the  difference  of  potential  at 
the  extremities  of  the  conductor.  Thus,  if  E  =  e.m.f.,  K  = 

TTl 

resistance  and  I  =  current,  I  =  -=-.     The  quantity  R  has 

it 

been  experimentally  proved  to  be  independent  of  the 
strength  of  the  current  (the  temperature  being  the  same) 
to  at  least  1  part  in  100,000. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  I,  art.  241.  Also  Thomson,  Elements  of  Elec- 
tricity and  Magnetism,  p.  284.  Also  Northrup,  Methods 
of  Measuring  Electrical  Resistance,  art.  105,  p.  16.) 

RESISTANCES  IN  SERIES. 

When  several  resistances  are  joined  in  series  the  total 
resistance  is  equal  to  the  sum  of  the  individual  resistances. 

From  coils  which  have  resistance-values  1-1-4-3  or 
1-3-3-2  (or  any  multiple  of  these  values)  the  successive 
values  0  to  9  can  be  obtained  by  moving  a  single  plug- 
connector. 

(Northrup,  Methods  of  Measuring  Electrical  Resistance, 
art.  503,  p.  82.) 


130  LAWS  OF  PHYSICAL_SCIENCE 

RESISTANCE    OF   A   NUMBER   OF    CONDUCTORS   ARRANGED 
IN  PARALLEL. 

When  a  number  of  resistances,  B±,  R>,  R3,  etc.,  are  joined 
together  in  parallel  combination  the  reciprocal  of  the  total 
resistance  R  is  equal  to  the  sum  of  the  reciprocals  of  the 
individual  resistances. 


Or,  calling  the  reciprocal  of  any  resistance  a  conductance, 
the  total  conductance  =  the  sum  of  the  conductances  of 
individual  conductors  when  joined  in  parallel  combination. 

If  n  resistance-units  are  used  singly,  and  joined  in  all 
possible  series  combinations,  2n  -  1  resistance-combinations 
can  be  obtained.  The  total  number  N  of  combinations 
possible  for  n  units  used  singly  and  joined  in  series  and 
parallel  combinations  is, 

N  =  2n+i  -  (n+2). 

(Ganot's  Physics,  art.  853.  Also  Northrup,  Methods  of 
Measuring  Electrical  Resistance,  art.  501,  p.  80.) 

CONDITION   FOR    A    DEFINITE    RESISTANCE. 

The  conductor  must  be  considered  as  having  its  surface 
divided  into  three  portions: 

1.  a  portion  over  which  the  potential  is  maintained  constant, 

2.  a  portion  over  which  the  potential  is  held  constant  but 

higher  or  lower  than  the  first  and 

3.  a  remaining  portion,  which  is  impervious  to  electricity. 
Only  when  the  conductor  is  in  approximately  the  above 

condition  can  its  resistance  be  said  to  be  definite. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  306.) 

RESISTANCE   OF  A  CONDUCTOR   CHANGES  WHEN  SPECIFIC 
RESISTANCE    CHANGES. 

If  the  specific  resistance  of  any  portion  of  a  conductor 
is  increased,  that  of  the  remainder  being  unchanged,  the 


ELECTRICITY  AND  MAGNETISM  131 

resistance  of  the  whole  conductor  will  be  increased,  and  if 
the  specific  resistance  of  any  portion  of  it  is  decreased,  that 
of  the  remainder  being  unchanged,  the  resistance  of  the 
whole  conductor  will  be  decreased. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  306.) 

FOUR-TERMINAL    CONDUCTORS. 

"The  generalized  four- terminal  conductor  is  a  mass  of 
conducting  material  of  any  size  or  shape  and  has  four 
limited  portions  of  the  surface  arbitrarily  selected  and 
adapted  for  making  electrical  connection  to  other  con- 
ductors. ' ' 

This  definition  (given  by  Frank  Wenner,  Bulletin  of  the 
Bureau  of  Standards,  Vol.  8,  p.  560)  covers  those  standards 
of  resistance  having  two  current  and  two  potential  ter- 
minals, which  are  used  in  fall  of  potential  methods  and  with 
the  Kelvin-double  bridge,  in  the  measurement  of  resistance. 

KIRCHHOFF'S   THEOREM   ON  INTERCHANGE   OF   ELECTRODES. 

* '  In  any  conductor  or  system  of  conductors  having  four 
terminals,  1,  2,  3,  and  4  selected  in  any  way,  the  drop  in 
potential  from  1  to  2  caused  by  a  current  entering  at  3 
and  leaving  at  4,  is  equal  to  the  drop  in  potential  from  3 
to  4  caused  by  an  equal  current  entering  at  1  and  leaving 
at  2." 

This  theorem,  given  by  Kirchhoff  in  1847,  is  of  impor- 
tance in  connection  with  the  measurement  of  resistances 
by  fall  of  potential  methods. 

(See  Bulletin  of  the  Bureau  of  Standards,  Vol.  VIII, 
p.  563.) 


132  LAWS  OF  PHYSICAL  SCIENCE 

JOULE'S  LAW  OF  GENERATION  OF  HEAT  IN  A  CONDUCTOR. 

The  heat  produced  by  the  passage  of  an  electric  current 
through  a  solid  metallic  conductor  is  proportional  to  the 
product  of  the  resistance  of  the  conductor,  the  square  of 
the  current  and  the  time,  or  to  the  product  of  the  applied 
e.m.f .  the  current  and  the  time. 

Thus,  JH  =  RPt  =  EIt, 

where  J  is  Joule's  dynamical  equivalent  of  heat,  H  the 
number  of  units  of  heat,  E  the  resistance  of  the  conductor, 
I  the  current,  t  the  time  during  which  the  current  flows  and 
E  the  applied  e.m.f.  IVhen  practical  units  are  used, 
H  =  0.24  RPt  in  gram-calories,  approximately. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  I,  art.  242.  Also  "  High  Temperature  Investiga- 
tion and  a  Study  of  Metallic  Conduction,'*  by  E'.  F. 
Northrup,  Jour,  of  the  Franklin  Institute,  June,  1915,  pp. 
650-652.) 

CONVERSION   OF  MECHANICAL   ENERGY  INTO  HEAT 
IN    A    CONDUCTOR. 

The  mechanical  work  done  by  electromotive  force  in 
driving  electricity  through  a  solid  metallic  conductor  is 
entirely  converted  into  heat.  If,  however,  the  metallic  con- 
ductor is  liquid  (molten)  some  power  is  spent  in  circulating 
the  liquid  metal. 

(Consult  paper  by  E.  F.  Northrup,  "A  New  Type  of 
Ammeter,  etc."  Proc.  of  the  American  Electrochemical 
Society,  May  7,  1909,  pp.  303-329.) 

MINIMUM  HEAT  CONDITION. 

"In  any  system  of  conductors  in  which  there  are  no 
internal  electromotive  forces  the  heat  generated  by  currents 
distributed  in  accordance  with  Ohm's  law  is  less  than  if  the 
currents  had  been  distributed  in  any  other  manner  con- 


ELECTRICITY  AND  MAGNETISM  133 

sistent  with  the  actual  conditions  of  supply  and  outflow  of 
the  current. ' ' 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  284.  Also  Thomson,  Elements  of  Electricity  and 
Magnetism,  pp.  310-313.) 

THE  WIEDEMANN-FRANZ  RATIO. 

The  best  conductors  of  electricity  are  also  the  best  con- 
ductors of  heat.  The  ratio  of  the  thermal  to  the  electrical 
conductivity  for  all  good  conductors  at  the  same  temperature 
has  the  same  value ;  the  value  of  this  ratio  is  proportional 
to  the  absolute  temperature.  This  law  holds  well  for  pure 
metals,  less  accurately  for  alloys  and  not  at  all  for  poor 
conductors. 

(Campbell,  Modern  Electrical  Theory,  pp.  66-68.  For 
experimental  illustrations  see  Richardson,  The  Electron 
Theory 'of  Matter,  pp.  410-413.) 

SUPERCONDUCTIVITY. 

When  the  temperature  of  a  very  pure  metal  is  reduced 
to  near  the  absolute  zero  of  temperature  (to  less  than  4° 
K.)  its  electrical  resistance  vanishes  or  its  conductivity 
becomes  practically  infinite. 

This  is  an  experimental  result  of  the  researches  of 
Kamerlingh  Onnes  and  has  been  termed  "Superconduc- 
tivity." 

(Consult,  "Electrical  Conductivity  at  High  Tempera- 
tures and  Its  Measurement,"  by  E.  F.  Northrup,  Trans,  of 
the  Amer.  Electrochem.  Soc.,  Vol.  XXV,  1914,  p.  377.  For 
data  see  Smithsonian  Physical  Tables,  p.  280.) 


134  LAWS  OF  PHYSICAL  SCIENCE 

RESISTANCE-TEMPERATURE  RELATIONS  FOR  METALS. 

The  variation  in  electrical  resistance  per  unit  of  resis- 
tance per  degree  is  called  the  "resistance-temperature 
coefficient"  of  a  substance.  If  Rt  is  the  ohmic  resistance  of 
a  sample  of  the  substance  when  at  temperature  t  then  its 
resistance-temperature  coefficient  at  this  temperature  is 

1     dRt 


A  few  general  relations  between  resistance  and  tempera- 
ture for  several  pure  metals  and  some  alloys  have  been 
obtained  experimentally.  Thus: 

1.  All  pure  metals  when  in  the  solid  state  increase  in  resist- 

ance  when   the    temperature    is   increased   and   the 
coefficient  at  is  approximately  the  same  for  all  pure 
metals  and  has  the  same  order  of  magnitude  as  the 
coefficients  of  expansion  of  gases. 
(Northrup,  Jour,  of  the  Franklin  Institute,  June,  1915, 
p.  636  et  seq.) 

2.  Several  pure  metals  (Ag,  Au,  Cu,  Pb,  Al,  Sn,  Sb,  Bi) 

when  in  the  molten  state  increases  in  resistance  linearly 
with  the  temperature,  namely,  the  coefficient  at  is 
strictly  a  constant  over  a  considerable  range  of  tem- 
perature. The  same  is  true  for  some  alloys  ;  Sn  +  Bi 
in  particular. 

(Northrup,  Jour,  of  the  Franklin  Institute,  June,  1915, 
Figs.  1  and  2,  pp.  638,  639.) 

3.  If  we  call  y^y—r^-  the  coefficient  of  volume-expansion  of 

a  molten  metal,  then  the  ratio  -y-  is  nearly  the  same 

quantity  (lying  within  the  limits  3.48  and  5.19)  for  at 
least  six  pure  metals  (Na,  K,  Sn,  Hg,  Pb,  Bi). 
(Northrup,  Trans,  of  the  Amer.  Electrochem.  Soc.,  Vol. 
XXV,  1914,  p.  388) 


ELECTRICITY  AND  MAGNETISM  135 

4.  The  resistance  of  most  metals  (Na,  Al,  K,  Cu,  Zn,  Cd,  Sn, 

Sb,  Bi,  Au,  Hg,  Pb,  have  been  studied)  upon  changing 
from  the  solid  to  the  molten  state  approximately 
doubles.  Antimony  and  bismuth  are  exceptions  to  the 
general  rule,  the  resistance  of  these  decreasing  when 
fusion  occurs. 
(See  reference  under  3) 

5.  The  change  in  the  resistivity  per  degree  C.  of  a  sample  of 

copper  is  0.00681  microhm  per  centimeter  cube,  or  the 
conductivity  of  copper  is  strictly  proportional,  over 
ordinary  ranges  of  temperature  (as  10°  to  100°C.), 
to  its  resistance-temperature  coefficient.  Thus  either 
may  be  deduced  from  knowledge  of  the  other. 
(J.  H.  Dellinger,  "  The  Temperature  Coefficient  of 

Resistance  of  Copper, "  Bulletin  of  the  Bureau  of  Standards, 

Vol.  7,  pp.  83,  84.) 

6.  When  the  resistivity  is  given,  at  any  temperature,  for 

pure  tin  and  pure  bismuth — both  being  in  the  molten 
state — then   the   resistivity   of   any   molten   alloy   of 
known  proportions  of  these  two  metals  may  be  calcu- 
lated,  because   the   resistivity   of  the   alloy  bears   a 
strictly  linear  relation  to  the  percentage  of  gram-atoms 
in  which  either  constituent  is  present. 
(Northrup   and  R.   G.   Sherwood,   "New  Methods  for 
Measuring  Resistivity  of  Molten  Materials ;  Results  for  Cer- 
tain Alloys,"  Jour,  of  the  Franklin  Institute,  Aug.,  1916.) 

7.  At  temperatures  which  exceed  about  1452°  C.,  the  melt- 

ing-point of  nickel,  all  known  substances  in  either  the 
solid  or  liquid  state  are  more  or  less  electrically  con- 
ducting and  above  this  temperature  it  is  impossible  by 
any  means  to  obtain,  even  approximately,  good  elec- 
trical insulation. 

(Northrup,  Jour,  of  the  Franklin  Institute,  March,  1915, 
p.  352.) 


136  LAWS  OF  PHYSICAL  SCIENCE 

A  GENERAL   RELATION   BETWEEN   CONDUCTANCE   AND   CAPACITY. 

Let  two  perfect  conductors  which  serve  as  electrodes  be 
immersed  in  an  electrically  conducting  homogeneous  medium 
of  electrical  conductivity  a  and  call  G  the  electrical  con- 
ductance between  the  two  electrodes.  Then  substitute  for 
the  electrically  conducting  medium  a  dielectric  medium  of 
specific  inductive  capacity  K  and  call  C  the  electrostatic 
capacity  between  the  same  two  electrodes.  Under  these 
circumstances  the  general  relation  holds 

G   =  4^ 
C   "   K  ' 

(Northrup,  "  Use  of  Analogy  in  Viewing  Physical  Phe- 
nomena," Jour,  of  the  Franklin  Institute,  July,  1908,  pp. 
31,  32.  Also  consult  Jeans,  Electricity  and  Magnetism,  pp. 
339,  340.) 

A   RELATION  BETWEEN    CAPACITY  AND   RESISTANCE. 

In  every  portion  of  an  electric  circuit  where  electrostatic 
lines  and  lines  of  current-flow  pass  through  the  same 
medium,  the  product  of  the  resistance  and  the  capacity, 

however  dimensions  are  varied,  is  constant  and  equal  to  ^. 
Here  p  is  the  resistivity  of  the  medium  and  K  its  specific 
inductive  capacity.  This  proposition  is  of  importance  in  the 
electrical  measurement  of  high  resistances. 

(Northrup,  Methods  of  Measuring  Electrical  Resistance, 
art.  902,  pp.  186-189.) 

ELECTROMOTIVE  FORCES  IN  SERIES. 

Electromotive  forces  add  in  series  as  scalar  quantities 
but  obey  the  law  of  algebraic  signs. 

Thus,  E  =  Cl  +  e2  +  e3+  (-e4)  +  (-e6)  +e6  + . 

(Consult  Jeans,  Electricity  and  Magnetism,  pp.  298, 
299.) 


ELECTRICITY  AND  MAGNETISM  137 

CONTACT   ELECTRICITY. 

When  two  different  metals  are  in  contact  there  is  in 
general  an  e.m.f.  acting  from  one  to  the  other.  Let  C  be 
taken  as  a  standard  metal :  then  if  the  potential  of  a  metal 
I  in  contact  with  C  at  zero  potential  is  i  and  that  of  a  metal 
Z  in  contact  with  C  at  zero  potential  is  z,  the  potential  of 
Z  in  contact' with  I  at  zero  potential  is  z  -  i. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  arts.  246-249.) 

LAW  OF  VOLTA. 

When  any  number  of  conductors,  which  conduct  elec- 
tricity without  electrolytic  dissociation,  are  joined  together 
to  form  a  closed  chain,  and  all  are  at  the  same  temperature, 
the  total  electromotive  force,  or  sum  of  the  contact-differ- 
ences of  potential  at  the  surfaces  of  union  of  pairs  of 
elements,  is  zero. 

Thus,  for  the  three  elements  a,  b,  c,  which  form  a  closed 
chain, 

V    -I-  V    4-  V    —  O 

v  ab  ^     v  be     '      v  ca  —  ^  > 

where  Vab  is  the  contact-difference  of  potential  between  the 
pair  of  elements  a,  b,  and  Vbc,  Vca,  have  similar  meanings. 

This  result  is  known  as  Volta's  Law. 

(Consult  Jeans,  Electricity  and  Magnetism,  p.  298.  For 
a  very  full  treatment,  see  Chwolson,  Traite  de  Physique, 
Vol.  IV,  Part  10,  pp.  198-200.) 


138  LAWS  OF  PHYSICAL  SCIENCE 

KIRCHHOFF'S  LAWS. 

1.  The  algebraic  sum  of  the  currents  which  meet  at  any 

point  is  zero. 

2.  In  any  closed  circuit  the  algebraic  sum  of  the  products 

of  the  current  and  resistance  in  each  of  the  conductors 
in  the  circuit  is  equal  to  the  electromotive  force  in  the 
circuit. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
304,  305.  For  an  illustration  of  the  application  of  Kirch- 
hoff's  laws,  see  Northrup,  Methods  of  Measuring  Electrical 
Resistance,  art.  301,  pp.  44,  45.) 

STEINMETZ'S    EXTENSION   OF   KIRCHHOFF'S   LAWS   TO 
ALTERNATING    CURRENTS. 

(a)  "The  sum  of  all  the  e.m.fs  acting  in  a  closed  circuit 
equals  zero,  if  they  are  expressed  by  complex  quan- 
tities, and  if  the  resistance  and  reactance  e.m.f  .s  are 
also  considered  as  counter  e.m.f  .s. ' ' 

(&)   "The  sum  of  all  the  currents  flowing  towards  a  dis- 
tributing point  is  zero,  if  the  currents  are  expressed 
as  complex  quantities." 
(Steinmetz,  Alternating  Current  Phenomena,  art.  31, 

p.  40.) 

RESOLVED  ELECTROMOTIVE  FORCE  AND  CURRENT. 

(a)  "The  sum  of  the  components,  in  any  direction,  of  all 
e.m.f.s  in  a  closed  circuit,  equals  zero,  if  the  resist- 
ance and  reactance  are  considered  as  counter 
e.m.f.s." 

(&)  "The  sum  of  the  components,  in  any  direction,  of  all 
the  currents  flowing  to  a  distributing  point  equals 
zero/' 
(Joule's  law  and  the  energy-equation  do  not  give  a 

simple  expression  in  complex  quantities  because  power  is  a 

quantity  of  double  the  frequency  of  the  current  or  e.m.f. 

wave.) 


ELECTRICITY  AND  MAGNETISM  139 

(Steinmetz,  Alternating  Current  Phenomena,  art.  31, 
p.  41.) 

STEINMETZ'S  EXTENSION  OF  OHM'S  LAW  TO  ALTERNATING 
CURRENT. 

E  =  ZI,  I  =  I-,  Z  =  f  , 

where  E,  I  and  Z  are  e.m.f.,  current  and  impedance,  ex- 
pressed in  complex  quantities. 

(For  a  full  explanation  of  these  symbols  and  their 
relations,  consult  Steinmetz,  Alternating  Current  Phe- 
nomena, Chap.  V.  See  art.  30,  p.  40.) 

WORK  DONE  BY  ELECTROMOTIVE  FORCE. 

The  work  done  by  an  e.m.f.  is  measured  by  the  product 
of  the  e.m.f.  into  the  quantity  of  electricity  which  crosses 
a  section  of  the  conductor  under  the  action  of  the  e.m.f. 
This  work  is  the  same  as  the  work  done  by  an  ordinary  force 
and  both  are  measured  by  the  same  standards  or  units. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  570.) 

POWER  IN  A  CIRCUIT  WHEN  THE  CURRENT  IS  ALTERNATING. 

The  average  power  in  a  circuit  when  the  current  is 
alternating  (sine-waves  assumed)  is, 


where  E  is  the  effective  value  of  the  electromotive  force,  I 
the  effective  value  of  the  current  and  <p   the  phase  angle 
between  the  current  and  the  impressed  electromotive  force. 
The  instantaneous  power  in  a  circuit  is, 

P  =  ei, 

where  e  and  i  are  the  instantaneous  values  of  the  electro- 
motive force  and  the  current  respectively. 

(Steinmetz,  Alternating  Current  Phenomena,  art.  5,  p.  6. 
Also  consult  Christie,  Electrical  Engineering,  art.  80,  pp. 
113-118.) 


140  LAWS  OF  PHYSICAL  SCIENCE 

LAW  OF  ACTION  OF  AN   ELECTRIC   CURRENT   ON  A   MAGNET. 

When  current  passes  through  a  straight  wire,  a  straight 
magnetized  needle  held  above  the  wire  tends  to  place  itself 
at  right  angles  to  the  wire  and  to  the  perpendicular  let 
fall  from  the  center  of  the  magnetized  needle  to  the  wire. 
The  direction  of  the  current  in  the  wire  and  the  direction 
of  the  lines  of  magnetic  force  which  encircle  the  wire  are 
related  as  are  the  forward  thrust  and  rotation  of  a  right- 
handed  screw. 

(Consult  Ganot's  Physics,  art.  835.) 

ELECTRIC  CIRCUIT  AND  A  MAGNETIC  SHELL  COMPARED. 

"A  current  flowing  in  any  closed  circuit  produces  the 
same  magnetic  field  as  a  certain  magnetic  shell  known  as 
the  'equivalent  magnetic  shell.'  This  shell  may  be  taken 
to  be  any  shell  having  the  circuit  for  its  boundary,  its 
strength  being  uniform  and  proportional  to  that  of  the 
current. ' ' 

(See  Jeans,  Electricity  and  Magnetism,  p.  415.) 

AMPERE'S   LAW    FOR   THE    MAGNETIC    FIELD    DUE    TO   ANY 
CLOSED    LINEAR    CIRCUIT. 

At  any  point  P,  not  in  the  wire  of  a  closed  circuit  carry- 
ing an  electric  current,  the  magnetic  force  due  to  the  current 
can  be  derived  from  a  potential  Q  where  Q  =  a  constant  X 
the  current  X  the  solid  angle  subtended  by  the  circuit  at  P. 
When  electromagnetic  measure  is  used  the  constant  is  unity, 
and  Q  =  i  w,  i  being  the  current  flowing  round  the  circuit 
and  o>  the  solid  angle  subtended  by  the  circuit  at  the  point  P. 

(Thomson,  .Elements  of  Electricity  and  Magnetism,  p. 
325.) 

FORCE    ON   A   UNIT   POLE    EXTERIOR    TO   A   LINEAR    CONDUCTOR. 

The  force  exerted  by  a  linear  conductor,  carrying  a 
current  I  on  a  unit  magnetic  pole  exterior  to  the  con- 
ductor, is  perpendicular  to  the  plane  through  the  axis  of 
the  conductor  and  the  pole. 


ELECTRICITY  AND  MAGNETISM  141 

When  electromagnetic  units  are  used  this  force  is  equal 
to  twice  the  intensity  of  the  current  divided  by  the  perpen- 
dicular distance  r  from  the  pole  to  the  axis  of  the  conductor. 


Here,  T0  is  the  force  in  dynes  which  would  act  on  a  unit 
magnetic  pole,  or  it  is  the  intensity  of  the  magnetic  field, 
at  a  distance  r  from  the  axis  of  the  conductor. 

(Consult  "Some  Newly  Observed  Manifestations  of 
Forces  in  the  Interior  of  an  Electric  Conductor,"  by  E.  F. 
Northrup,  Physical  Review,  June,  1907,  p.  478.  Also  Max- 
well, Treatise  on  Electricity  ctmd  Magnetism,  arts?.  477-479.) 

MAGNETIC  FORCE  IN  THE  INTERIOR  OF  A  CONDUCTOR  OF 
CIRCULAR  CROSS-SECTION. 

The  lines  of  magnetic  force  form  circles  about  the  axis 
of  a  linear  conductor  of  circular  cross-section.  They  vanish 
at  the  axis  and  are  a  maximum  at  the  circumference  of  the 
conductor  and  increase  uniformly  with  the  distance  from 
axis  to  circumference. 

If  electromagnetic  units  are  used,  the  intensity  of  the 
magnetic  field  in  the  substance  of  the  conductor  is, 


where  I  is  the  current  in  the  conductor, 

R  the  radius  of  the  conductor  and 

r  the  distance  from  the  axis  to  where  the  intensity  of 
the  magnetic  force  is  Tj. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  II,  art.  683,) 


142  LAWS  OF  PHYSICAL  SCIENCE 

A    SMALL    ELECTRIC    CIRCUIT    COMPARED    WITH    A    MAGNET. 

"The  magnetic  action  of  a  small  plane-circuit  at  dis- 
tances which  are  great  compared  with  the  dimensions  of 
the  circuit  is  the  same  as  that  of  a  magnet  whose  axis  is 
normal  to  the  plane  of  the  circuit,  and  whose  magnetic 
moment  is  equal  to  the  area  of  the  circuit  multiplied  by 
the  strength  of  the  current.  " 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  482.) 

MAGNETIC   POLE   AND    EQUIPOTENTIAL    SURFACES. 

"The  force  acting  on  a  magnetic  pole  placed  at  any 
point  of  an  equipotential  surface  is  perpendicular  to  this 
surface,  and  varies  inversely  as  the  distance  between  con- 
secutive surfaces." 

(See  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  II,  art.  487.) 

LAW  OF  THE  INTERNAL  PRESSURE  PRODUCED  BY  AN  ELECTRIC 
CURRENT   IN   A   CONDUCTOR. 

In  every  conductor  which  carries  an  electric  current,  a 
pressure  is  produced  within  the  substance  of  the  conductor 
which  results  from  the  mutual  attractions  of  all  the  current- 
carrying  elements  of  the  conductor. 

When  the  conductor  has  a  circular  cross-section,  the 
pressure  is  directed  toward  its  axis.  Its  value  per  unit 
area  is  a  maximum  at  the  axis  and  decreases  to  zero  at  the 
circumference.  When  the  radius  of  the  conductor  is  R,  and 
the  current  it  carries  is  I,  the  pressure  g  per  unit  area  at 
any  radial  distance  r  from  the  axis  is, 


When  I  is  in  electromagnetic  measure  and  R  and  r  are  in 
centimeters  g  is  in  dynes  per  cm2. 

(See  "Some  Newly  Observed  Manifestations  of  Forces 


ELECTRICITY  AND  MAGNETISM  143 

in   the    Interior   of   an    Electric    Conductor/'   by   E.   F. 
Northrup,  Physical  Review,  June,  1907.) 

LONGITUDINAL   MOTION  IN  AN   ELECTRICAL    CONDUCTOR. 

When  by  any  geometrical  disposition  whatever  of  an 
electric  circuit,  in  which  the  conducting  material  is  a  fluid 
capable  of  free  motion,  normally  acting  electrodynamic 
forces  arise  in  any  section  of  the  circuit  which  vary  in 
magnitude  from  one  point  to  another  over  a  length  meas- 
ured along  the  axis  of  the  conductor,  there  also  arise 
hydrodynamic  forces  which  can  impress  motions  on  the 
fluid  substantially  parallel  to  the  longitudinal  axis  of  the 
conductor. 

(This  generalization  is  drawn  from  investigations  by 
E.  F.  Northrup  which  are  not  yet  published.) 

WORK   DONE   IN    MOVING   A   MAGNETIC   POLE    ROUND 
A  CLOSED  CURVE. 

If  there  exists  a  magnetic  field  due  to  electric  currents 
and  a  closed  curve  is  drawn  in  this  field,  the  work  done  in 
moving  a  magnetic  pole  of  strength  m  round  the  closed 
curve  is  zero  if  the  closed  curve  does  not  thread  an  electric 
circuit,  and  47rm  times  the  current  in  any  circuit  which  the 
closed  curve  threads  once. 

Thus,  W  =  47rlm  is  the  work  done  when  the  closed  curve 
threads  once  a  circuit  which  carries  a  current  I,  and  W±  = 
4-TrImn  when  the  closed  curve  threads  the  circuit  n  times. 
The  value  of  the  line-integral  4?rl  is  independent  of  the 
medium  in  which  the  closed  curve  is  drawn. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  arts.  480,  498,  499.) 


144  LAWS  OF  PHYSICAL  SCIENCE 

FUNDAMENTAL  LINE-INTEGRALS. 

The  two  propositions: 

1.  The   line-integral   of   the   magnetic   force    round   any 

closed  curve  is  equal  to  4?r  times  the.  total  current 
flowing  through  the  closed  curve  and 

2.  The  line-integral  of  the  electric  force  round  any  closed 

curve  is  equal  to  the  time-rate  of  diminution  of  the 
total  magnetic  induction  included  by  the  closed  curve, 
are  basic  in  many  mathematical  investigations  of 
alternating  currents. 

(For  application  of  these  principles,  see  "The  Skin 
Effects  and  Alternating  Current  Resistance, "  by  E.  F. 
Northrup  and  John  R.  Carson,  Jour,  of  the  Franklin  Insti- 
tute, Feb.,  1914,  p.  141  et  seq.) 

VECTORIAL   ADDITION  OF  MAGNETIC  AND  ELECTRIC  FORCES. 

Magnetic  and  electric  forces  or  intensities  in  a  homo- 
geneous medium  must  add  vectorially,  giving  a  resultant 
intensity. 

By  combining  this  principle  with  certain  elementary 
laws  of  electrostatics  and  magnetism,  a  large  number  of 
theorems  have  been  demonstrated. 

(Consult  Jeans,  Electricity  and  Magnetism,  pp.  26, 
358.) 

INTERACTION  OF  MAGNETS  AND  ELECTRIC  CURRENTS. 

The  mechanical  action  of  currents  on  magnets  is  equal 
and  opposite  to  the  action  of  magnets  on  currents. 

•  It  is  shown  by  theory  and  experiments  that  a  bar- 
magnet  magnetized  to  have  a  North  pole  at  each  end  and  a 
South  pole  at  its  center,  when  freely  suspended  in  a  hori- 
zontal position  in  a  vertical,  conducting,  liquid  column  of 
circular  cross-section  will  rotate  with  continuous  rotation 
when  the  liquid  carries  an  electric  current.  Reversing  the 
direction  of  the  current  reverses  the  direction  of  rotation. 


ELECTRICITY  AND  MAGNETISM  145 

(Ganot's  Physics,  arts.  888,  889.  Also  see  article  by 
E'.  F.  Northrup,  Physical  Review,  June,  1907,  p.  480.) 

ELECTRIC  AND  MAGNETIC  ANALOGIES. 

Electric  system. 

1.  The  line-integral  of  the  electric  force  round  any  closed 

curve  passing  through  the  battery  is  E,  while  round 
any  other  closed  curve  it  vanishes. 

2.  The  lines  of  flow  of  electric  current  are  closed  curves 

which  pass  through  the  battery. 

3.  The  density  of  the  electric  current  is  a  times  the  electric 

force,  where  o-  is  the   conductivity  of  the  medium 
carrying  the  current. 
Magnetic  system. 

1.  The  line-integral  of  the  magnetic  force  round  any  closed 

curve  which  threads  a  magnetizing  circuit  is  4;rl,  while 
round  any  other  closed  curve  it  vanishes. 

2.  The  lines  of  magnetic  induction  are  closed  curves  which 

thread  the  magnetizing  circuit. 

3.  The  magnetic  induction  is  /u,  times  the  magnetic  force, 

where  p  is  the  magnetic  permeability.   V^ 
(For  amplification  and  applications  of  these  analogies, 
see  Thomson,  Elements  of  Electricity  and  Magnetism,  pp, 
348-350.) 

LAW   OF   MAGNETIC   INDUCTION.    (l) 

"When  the  number  of  lines  of  magnetic  induction  which 
pass  through  the  secondary  circuit  in  the  positive  direction 
is  altered,  an  electromotive  force  acts  round  the  circuit, 
which  is  measured  by  the  rate  of  decrease  of  the  magnetic 
induction  through  the  circuit" — provided  the  integrity  of 
the  original  circuit  is  preserved. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  531). 


146  LAWS  OF  PHYSICAL  SCIENCE 

LAW   OF  MAGNETIC   INDUCTION,    (a) 

Another,  and  to  many,  a  preferable  statement  of  this  law 
is :  Whenever  a  real  or  imaginary  }ine  in  space  is  being  cut 
at  right  angles  to  itself  by  tubes  of  magnetic  induction  an 
e.m.f .  acts  along  this  line  which  is  proportional  to  the  rate 
of  cutting. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  II,  art.  541.) 

LENZ'S  LAW. 

When  a  circuit  is  moved  in  a  magnetic  field  in  such 
a  way  that  a  change  takes  place  in  the  number  of  tubes  of 
magnetic  induction  passing  through  the  circuit,  a  current 
is  induced  in  the  circuit  and  a  mechanical  force  is  set  up 
such  that  this  force  tends  to  stop  the  motion  which  gave 
rise  to  the  current. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
440.  Also  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  II,  art.  542.) 

LINE    OF    MAGNETIC    INDUCTION    DEFINED. 

Maxwell  defines,  negatively,  a  line  of  magnetic  induction 
in  four  ways: 

1.  If  a  conductor  be  moved  along  it  parallel  to  itself  it  will 

experience  no  electromotive  force. 

2.  If  a  conductor  carrying  a  current  be  free  to  move  along 

a  line  of  magnetic  induction  it  will  experience  no 
tendency  to  do  so. 

3.  If  a  linear  conductor  coincides  in  direction  with  a  line 

of  magnetic  induction  and  be  moved  parallel  to  itself 
in  any  direction,  it  will  experience  no  electromotive 
force  in  the  direction  of  its  length. 

4.  If  a  linear  conductor  carrying  an  electric  current  co- 

incide in  direction  with  a  line  of  magnetic  induction 
it  will  not  experience  any  mechanical  force. 


ELECTRICITY  AND  MAGNETISM  147 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  597.) 

ELECTROMOTIVE  FORCE  INDEPENDENT  OF  THE  NATURE 
OF  THE  CONDUCTOR. 

The  intensity  of  an  electromotive  force  which  results 
from  electromagnetic  induction  is  entirely  independent  of 
the  nature  of  the  substance  of  the  conductor  in  which  the 
electromotive  force  acts,  and  also  of  the  nature  of  the 
conductor  which  carries  the  inducing  current. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  534.) 

MECHANICAL  FORCE  ACTS  ON  THE  CONDUCTOR,  NOT  ON  THE 
CURRENT. 

The  mechanical  force  which  urges  a  conductor  carrying 
a  current  across  lines  of  magnetic  force,  acts,  not  on  the 
electric  current  but  on  the  conductor  which  carries  it. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  501.) 

MECHANICAL  FORCE  ACTING  ON  A  CONDUCTOR  REPRESENTED 
BY  A  PARALLELOGRAM. 

The  mechanical  force  which  acts  upon  unit  length  of  a 
conductor  carrying  a  current  is  numerically  equal  to  the 
area  of  a  parallelogram,  two  sides  of  which  are  drawn 
parallel  to  the  conductor  and  proportional  to  the  strength 
of  the  current  at  any  point,  the  other  two  sides  being  drawn 
parallel  and  proportional  to  the  magnetic  induction  at  the 
same  point.  The  mechanical  force  is  normal  to  the  plane 
of  the  parallelogram  so  drawn.  If  a  right-handed  screw 
be  turned  from  the  direction  of  the  current  to  the  direc- 
tion of  the  induction  the  direction  in  which  the  mechanical 
force  acts  coincides  with  the  direction  of  forward  motion 
of  the  screw. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  500.) 


148  LAWS  OF  PHYSICAL  SCIENCE 

MAGNETIC  ENERGY  IS  POTENTIAL  ENERGY. 

The  energy  of  any  strictly  magnetic  system  may  be 
considered  as  potential  energy  and  if  so  considered  this 
energy  is  always  diminished  when  the  parts  of  the  system 
yield  to  the  magnetic  forces  which  act  on  them. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  638.) 

MUTUAL   ACTION  BETWEEN  TWO   CIRCUITS   IS   DEPENDENT   UPON 
A  SINGLE   QUANTITY. 

All  phenomena  of  the  mutual  action  of  two  circuits, 
whether  the  induction  of  currents  or  the  mechanical  force 
which  acts  between  them,  depend  upon  the  value  of  the 
coefficient  of  mutual  induction  between  the  circuits,  a 
coefficient  which  depends  only  upon  the  geometrical  rela- 
tions of  the  circuits. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  584.  Very  exact  formulas  for  calculating  coefficients 
of  self  induction  and  mutual  induction  are  to  be  found  in  a 
series  of  papers  by  Dr.  E.  B.  Rosa  in  volumes  2,  3,  and  4 
of  the  Bulletin  of  the  Bureau  of  Standards.) 

VECTOR    RELATIONS    OF    VELOCITY,    INDUCTION    AND 
ELECTROMOTIVE   FORCE. 

''The  magnitude  of  the  electromotive  force  is  repre- 
sented by  the  area  of  the  parallelogram,  whose  sides  rep- 
resent the  velocity  and  the  magnetic  induction,  and  its 
direction  is  the  normal  to  this  parallelogram,  drawn  so  that 
the  velocity,  the  magnetic  induction  and  the  electromotive 
force  are  in  right-handed  cyclical  order." 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  599.  Also  see  art.  594,  where  is  shown  how  a  volume 
may  be  made  to  represent  the  increment  in  the  electro- 
kinetic  momentum  of  a  secondary  circuit.) 


ELECTRICITY  AND  MAGNETISM  149 

INDUCTION   COEFFICIENTS. 

In  every  electric  circuit  there  is  a  certain  quantity  L, 
called  the  coefficient  of  self  induction  of  the  circuit.  The 
magnitude  of  this  quantity  is  dependent  only  upon  the 
geometrical  dimensions  of  the  circuit.  If  the  currents  are 
constant,  changes  in  L  always  give  rise  to  e.m.f.s,  and  if  L 
is  constant  changes  in  the  currents  give  rise  to  e.m.f.s.  If 
two  or  more  circuits  are  in  the  neighborhood  of  each  other, 
there  is  for  each  of  the  circuits  a  self-induction  coefficient 
and  in  addition  other  quantities  M,  M.19  M2,  etc.,  called 
coefficients  of  mutual  induction  which  depend  only  upon 
the  geometrical  relations  of  the  circuits.  If  there  is  a 
steady  current  in  one  of  the  circuits,  and  there  are  changes 
in  L,  L1?  L2,  etc.,  or  in  M,  Mly  M2,  etc.,  e.m.f.s  are  always 
produced.  Or  if  these  quantities  are  constant,  changes  in 
the  current  produce  e.m.f.s. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  II,  arts.  540,  578,  579.) 

COIL    TO    GIVE    MAXIMUM    SELF    INDUCTION. 

When  the  weight  or  length  of  a  wire  is  given,  the  form 
in  which  to  wind  this  wire  in  a  channel  of  square  cross- 
section,  in  order  to  form  a  coil  of  maximum  self  induction, 
is  obtained  by  making  the  mean  radius  r  of  the  coil  equal 
1.85  times  a  side  of  the  section  of  the  channel. 

The  self  induction  in  henrys  is  then, 
L  =  19.347rn2  ICh9, 
where  n  is  the  number  of  turns  of  wire. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  706.  Also  Bulletin  of  the  Bureau  of  Standards, 
Vol.  2,  p.  108.) 


150  LAWS  OF  PHYSICAL  SCIENCE 

EXPRESSION   FOR   KINETIC    ENERGY   OF   TWO   CIRCUITS. 

The  kinetic  energy  of  a  system  formed  of  two  circuits 
when  currents  in  the  first  circuit  induce  currents  in  the 
second  circuit  is  given  by  the  expression, 

T  =  y2I2lL  +  I1IM  +  y2l22N) 

where  L  is  the  coefficient  of  self  induction  of  first  circuit 
and  N  of  the  second  circuit,  and  M  is  the  coefficient  of 
mutual  induction  between  the  two  circuits,  and  I±  and  I2 
are  currents  in  the  first  and  second  circuits  respectively. 
(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  581.) 

MAGNETIC   ENERGY   COMPARED    WITH   ELECTROKINETIC    ENERGY. 

It  is  always  possible  to  make  an  arrangement  of  infi- 
nitely small  electric  circuits  which  shall  correspond  in  all 
respects  to  any  magnetic  system,  provided  that  in  calcu- 
lating the  potential  we  avoid  passing  through  any  of  these 
small  electric  circuits  with  a  line  of  integration. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  637.) 

VECTOR-POTENTIAL. 

The  vector-potential  is  a  quantity  which  represents  in 
direction  and  magnitude,  the  time-integral  of  the  intensity 
of  the  electromotive  force  which  a  particle  placed  at  a 
point  in  a  magnetic  field  would  experience  if  the  current 
to  which  the  magnetic  field  is  due  were  suddenly  stopped. 
It  is  identical  with  the  electrokinetic  momentum  at  the 
point. 

All  lines  of  magnetic  induction  through  a  closed  curve 
in  being  removed  are  shown  by  Maxwell  to  equal  the  line- 
integral  of  the  resolved  part  of  the  vector-potential  taken 
round  the  curve.  This  line-integral,  physically  interpreted, 
is  the  total  electrokinetic  momentum  of  the  closed  curve 
or  circuit,  and  when  the  line-integral  is  extended  round  a 


ELECTRICITY  AND  MAGNETISM  151 

primary  circuit  it  is  numerically  equal  to  the  product  of 
the  self  induction  of  the  circuit  by  the  current  in  the  circuit. 
It  is  the  analogue  of  mass  X  velocity,  or  momentum  in 
mechanics. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  arts.  405,  590,  592.) 

ELECTROMOTIVE  FORCE  IMPRESSED  ON  A  CIRCUIT. 

If  an  e.m.f.  is  momentarily  impressed  on  any  electric 
circuit  three  cases  may  exist: 

1.  The  movement  of  electricity  is  retarded  by  an  ohmic 

resistance  only. 

2.  It   is   retarded   by   ohmic   resistance   and  opposed   by 

magnetic  inertia. 

3.  It   is   retarded   by   ohmic   resistance    and   opposed   by 

magnetic  inertia  and  by  a  counter  e.m.f.  which  varies 
as  a  function  of  the  impressed  e.m.f. 
In  the  3rd  case  when  the  damping  by  ohmic  resistance 
is  negligible  the  electricity  always  tends  to  oscillate  upon 
the  sudden  removal  of  the  impressed  e.m.f.  and  the  period 
is  T  =  2fr  V  LC,  where  L  is  the  self  induction  and  C  the 
capacity  of  the  circuit. 

(Consult  Bedell  and  Crehore,  Alternating  Currents, 
Part  1.) 

AN  ELECTROMOTIVE  FORCE  ACTS  ONLY  ON  ELECTRICITY. 

An  electromotive  force  has  of  itself  no  tendency  to 
cause  the  mechanical  motion  of  any  body,  but  acts  only  to 
cause  a  movement  of  electricity. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  art.  535.) 


152  LAWS  OF  PHYSICAL  SCIENCE 

DECAY  OF  INTERNAL  CHARGES  IN  DIELECTRICS. 

If  in  the  interior  of  a  mass  of  homogeneous  poorly  con- 
ducting material,  there  exists  at  any  point  an  electric  charge 
it  will  tend  to  die  away.  Neither  its  formation  nor  the  rate 
at  which  it  dies  away  is  influenced  by  the  application  of 
external  e.m.f.s  which  do  not  lead  to  disruptive  discharges. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 

I,  art.  325.) 

GENERAL  PRINCIPLE  OF  MECHANICAL  ACTION  OF  CURRENTS. 

All  mechanical  actions  of  electric  currents  depend  upon 
the  strength  of  the  currents  and  not  upon  their  rate  of 
variation  and  all  mechanical  actions  of  currents  remain  the 
same  when  all  the  currents  are  reversed  in  direction. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 

II,  art.  574.) 

MUTUAL  RELATIONS   OF   CURRENTS. 

1.  If  a  current  traverses  a  wire  and  returns  by  a  tube 

surrounding  the  wire  there  is  no  magnetic  field  ex- 
ternal to  the  tube. 

2.  The  external  action  of  a  crooked  wire  (bent  like  a  row 

of  saw-teeth)  upon  a  neighboring  wire  is  the  same  as 
that  of  a  straight  wire. 

3.  If  a  wire  carries  a  current  no  external  magnetic  force 

can  so  act  upon  the  wire  as  to  tend  to  make  it  move 
in  the  direction  of  its  length. 

4.  The  force  acting  between  two  elements  of  two  electric 

circuits  is  inversely  proportional  to  the  square  of  the 
distance  between  them. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
II,  arts.  505-508.) 

INTERACTION    OF    ELECTRIC    CONDUCTORS. 

1.  Two  conductors  which  are  parallel  and  carry  currents 
in  the  same  direction  attract  one  another,  and  when 


ELECTRICITY  AND  MAGNETISM  153 

they  carry  currents  in  opposite  directions  they  repel 
one  another. 

2.  Two  rectilinear  current-carrying  conductors,  when  their 
directions  are  such  that  they  form  an  angle  with  each 
other,  attract  one  another  if  the  currents  in  both  con- 
ductors approach  or  recede  from  the  apex  of  the  angle, 
and  they  repel  one  another  if  one  current  approaches 
and  the  other  recedes  from  the  apex  of  the  angle. 
(Ganot's  Physics,  arts.  880-882.) 

LAW  OF  RESOLUTION  OF  CURRENTS. 

The  law  of  the  resolution  of  electric  currents  is  the 
same  as  that  of  velocities,  forces  and  all  other  vectors. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol. 
I,  art.  286.) 

MAGNETOMOTIVE   FORCE,  AND   MAGNETIC   INDUCTION 
IN  A   SOLENOID. 

In  a  solenoid  of  infinite  length  having  n  turns  per  unit 
length,  the  magnetic  force  is  uniform  over  the  cross-section 
of  the  solenoid  and  its  value  is  H  =  47rni,  where  i  is  the 
current  in  electromagnetic  measure.  Or,  if  I  is  the  current 
in  amperes  the  magnetomotive  force  is 

H  =  1.2566nl  gilberts. 
The  induction  per  cm2  in  the  solenoid  is, 

B  =  1.2566nl/>i  gausses,  where  /*  is  the  permeability 
of  the  medium  within  the  solenoid. 

(Consult  Christie,  Electrical  Engineering,  p.  58  et 
seq.) 


154  LAWS  OF  PHYSICAL  SCIENCE 

A  FUNDAMENTAL   ENGINEERING  EQUATION. 

The  following  equation  is  of  fundamental  importance  in 
electrical  engineering : 

Eeff  =  V~2Vn  ^NIO-8  =  4.44n  «?  N10-8  volts. 

Here,  Eeff  =  effective  e.m.f.,  n  =  total  number  of  turns  in 
the  circuit,  <p  =  the  total  maximum  flux  through  the 
circuit  and  N  =  the  frequency  or  number  of  complete 
cycles  per  second  of  the  magnetizing  current,  sine  waves 
being  assumed. 

(Steinmetz,  Alternating  Current  Phenomena,  Chap.  III. 
See  p.  17.) 

THERMOELECTRIC    CURRENTS. 

If  an  electric  circuit  is  made  up  of  two  unlike  metals  and 
one  junction  of  the  two  metals  is  maintained  at  a  higher 
temperature  than  the  other  junction,  an  electric  current  will 
flow  in  the  circuit,  and  as  the  result  of  this  current  heat  will 
be  transferred  from  the  hotter  toward  the  colder  junction. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
501.  Also  Chwolson,  Traite  de  Physique,  Vol.  IV,  Part  10, 
p.  737.) 

THERMOELECTRIC  LAW,  THERMOELECTRIC  POWER. 

If  et  is  the  e.m.f.  of  a  bimetallic  circuit  when  the  cold 
junction  is  at  temperature  t0  and  the  hot  Junction  at  ^  and 
if  e2  is  the  e.m.f.  when  the  cold  junction  is  at  tx  and  the  hot 
junction  at  t2,  then,  when  the  cold  junction  is  at  t0  and  the 
hot  junction  is  at  t2  the  e.m.f.  is  e^  4-  e2.  Or,  the  e.m.f.  E 
round  a  circuit  whose  junctions  are  at  temperatures  tt  and 

tg  is  E  =/t*  Qdt,  where  Qdt  is  the  e.m.f.  round  the  circuit 
when  the  temperature  of  the  cold  junction  is  t>-  %  dt  and 
the  temperature  of  the  hot  Junction  is  t  -f  %  dt.  The  quan- 
tity Q  is  known  as  the  thermoelectric  power  of  the  circuit 
at  temperature  t. 


ELECTRICITY  AND  MAGNETISM  155 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
506.  Also  consult  Chwolson,  Traite  de  Physique,  Vol.  IV, 
Part  10,  p.  744.  For  values  of  thermoelectric  power  see 
Smithsonian  Physical  Tables,  pp.  268-270.) 

THERMOELECTRIC  LAW  FOR  DIFFERENT  PAIRS  OF  METALS. 

If  Eac  is  the  e.m.f .  round  a  circuit  formed  of  the  pair  of 
metals  A,  C,  and  Ebc  the  e.m.f.  round  a  circuit  formed  of  the 
pair  of  metals  B,  C,  then  Eac  -  Ebc  is  the  e.m.f.  round  a 
circuit  formed  of  the  metals  A  and  B;  all  the&e  circuits 
being  supposed  to  work  between  the  same  limits  of 
temperature. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
506.) 

PELTIER  EFFECT. 

When  current  flows  across  the  junction  of  two  unlike 
metals  it  gives  rise  to  an  absorption  or  liberation  of  heat. 
If  the  current  flows  in  the  same  direction  as  the  current  at 
the  hot  junction  in  a  thermoelectric  circuit  of  the  two  metals 
heat  is  absorbed;  if  it  flows  in  the  same  direction  as  the 
current  at  the  cold  junction  of  the  thermoelectric  circuit 
heat  is  liberated.  This  phenomenon  is  known  as  the  Peltier 
Effect. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  pp. 
501,  502.  Also  Ganot's  Physics,  art.  878.  Consult  also 
Smithsonian  Physical  Tables,  p.  271.) 


156  LAWS  OF  PHYSICAL  SCIENCE 

MEASURE   OF  PELTIER  EFFECT. 

The  Peltier  Effect  equals  (the  thermoelectric  power) 
X  (  the  absolute  temperature).  Or,  P  =  QT,  where  T  = 
absolute  temperature. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
510.  Also  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  1,  art.  249.) 

The  Peltier  Effect,  the  closely  associated  Thomson  Effect 
and  the  Hall  Effect  are  thought  by  many  to  be  explained 
with  fair  satisfaction  on  the  modern  electron-theory.  For 
advanced  views  on  this  matter  consult  Campbell;  Modern 
Electrical  Theory,  pp.  71-74.) 

ACTION  OF  CURRENT  FLOWING  ALONG  AN  UNEQUALLY  HEATED 
CONDUCTOR,  CALLED  THE  "  THOMSON  EFFECT." 

When  an  electric  current  flows  along  an  unequally 
heated  metallic  conductor  it  tends,  in  the  case  of  copper,  to 
diminish  the  inequality  of  temperature  and  in  the  case  of 
iron  to  increase  this  inequality. 

(Thomson,  Elements  of  Electricity  and  Magnetism,  p. 
505.  Also,  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  I,  art.  253.  Also,  for  full  account,  see  Chwolson,  Trait  e 
de  Physique,  Vol.  IV,  Part  10,  pp.  752-756.) 

RICHARDSON'S   LAW   OF   ELECTRONIC    EMISSION. 

*  'The  number  of  electrons  emitted  at  different  tempera- 
tures T  is  governed  by  the  formula, 

N  =  ATVb/T. 

A,  X  and  b  are  constants/'  A  varies  greatly  with  the  sub- 
stance, A.  is  not  far  different  than  unity  and  b  in  equivalent 
volts  is  always  comparable  with  five. 

(Richardson,  The  Electron  Theory  of  Matter,  p.  441. 
Also  0.  W.  Richardson,  Phil.  Trans.  (A),  Vol.  CGI,  p.  543, 
1903.) 


ELECTRICITY  AND  MAGNETISM  157 

LAW    OF    MAGNUS. 

If  a  circuit  is  formed  of  a  single  metal  no  current  will 
be  formed  in  it  however  the  section  of  the  conductor  and 
the  temperature  may  vary  in  different  parts  of  the  circuit. 

(Maxwell,  Treatise  on  Electricity  and  Magnetism,  Vol.  I, 
art.  251.) 

THERMOELECTRIC    INVERSION. 

Call  E  the  e.m.f.  acting  round  a  thermoelectric  circuit 
of  two  metals.  Then,  when  the  difference  of  temperature 
of  the  two  junctions  is  twice  the  difference  of  temperature 
for  which  E  is  a  maximum,  E  becomes  zero  and  a  further 
increase  in  the  temperature-difference,  the  lower  tempera- 
ture of  the  one  junction  being  held  constant,  causes  E  to 
change  direction.  This  is  known  as  thermoelectric  inversion. 

(See  Chwolson,  Trait  e  de  Physique,  Vol.  IV,  Part  10,  p. 
739.  Thermoelectric  phenomena  are  very  adequately  and 
fully  treated  in  Chap.  VI,  pp.  728-761.) 

RATIO  OF  THE  ELECTROMAGNETIC  TO  THE  ELECTROSTATIC  UNIT 
OF  A  QUANTITY  OF  ELECTRICITY. 

The  ratio  of  the  electromagnetic  to  the  electrostatic  unit 
of  a  quantity  of  electricity  is  a  velocity  (whatever  units  of 
length  and  time  are  chosen)  and  this  velocity  is  experi- 
mentally shown  to  be  the  velocity  of  light.  The  ratio  is 
called  v  and,  very  nearly, 

v  =  3  X  1010'  cm/sec. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism,  Vol.  II,  Chap.  XIX.  Also  Thomson,  Elements  of 
Electricity  and  Magnetism,  p.  470  et  seq.) 


158  LAWS  OF  PHYSICAL  SCIENCE 

RELATION  OF  SPECIFIC  INDUCTIVE  CAPACITY  AND  INDEX 
OF  REFRACTION. 

According  to  Maxwell's  electromagnetic  theory  of  light, 
the  dielectric  capacity  (specific  inductive  capacity)  of  a 
transparent  medium  is  equal  to  the  square  of  its  index  of 
refraction  for  electromagnetic  radiation. 

Thus,  K  =  p.2. 

(Consult  Maxwell,  Treatise  on  Electricity  and  Magnet- 
ism, Vol.  II,  arts.  788,  789.) 

ROTATION   OF  THE   PLANE   OF  POLARIZATION   OF  LIGHT. 

In  1845  Faraday  discovered  that  if  a  substance  which 
ordinarily  will  not  rotate  plane-polarized  light  be  placed  in 
a  strong  magnetic  field  it  acquires  this  property.  In  looking 
from  North  to  South  along  a  line  of  magnetic  force  the 
rotation  is  clockwise.  On  reversing  the  direction  of  magneti- 
zation, the  direction  of  rotation  is  reversed. 

The  angular  rotation  of  the  plane  of  polarization  of  a 
plane-polarized  ray  of  light,  which  is  parallel  to  a  magnetic 
field,  is  numerically  equal  to  the  amount  by  which  the 
magnetic  potential  increases  in  passing  from  the  point  where 
the  ray  enters  the  medium,  (in  which  the  rotation  takes 
place)  to  the  point  where  the  ray  leaves  it,  multiplied  by  a 
coefficient.  This  coefficient  is  generally  positive  for  dia- 
magnetic  media. 

(See  Preston,  The  Theory  of  Light,  p.  431.  Also  Max- 
well, Treatise  on  Electricity  and  Magnetism,  Vol.  II,  art. 
808.  For  formulae  and  numerical  values  of  Verdet's  con- 
stant see  Smithsonian  Physical  Tables,  pp.  326-330.) 

PRESSURE  OF  RADIANT  ENERGY. 

In  a  medium,  in  which  radiant  energy  is  propagated  as 
light,  heat  or  electromagnetic  waves,  there  is  a  pressure  in 
the  direction  normal  to  the  wave  front.  This  pressure  is 
numerically  equal  to  the  energy  in  unit  volume  of  the 
medium. 


ELECTRICITY  AND  MAGNETISM  159 

If  K  is  the  specific  inductive  capacity  and  /x,  the  permea- 
bility of  the  medium,  P  the  maximum  electromotive  force 
and  ft  the  maximum  magnetic  force  (the  direction  of  which 
is  at  right  angles  to  both  the  direction  of  P  and  the  direc- 
tion of  the  propagation  of  the  wave)  then, 

•rr 

-g^-P2  =  ^-/32=  mean  energy  in  unit  volume  of  the 
medium. 

(See  Maxwell,  Treatise  on  Electricity  and  Magnetism, 
Vol.  II,  arts.  792,  793.  For  description  of  Nichols  and 
Hull's  experimental  proof  of  the  above,  see  Wood,  Physical 
Optics,  p.  466.) 

HALL   EFFECT. 

When  a  thin  rectangular  sheet  of  metal  carrying  an 
electric  current  flowing  in  the  direction  of  its  length  is  sub- 
jected to  a  powerful  magnetic  field  normal  to  the  sheet,  the 
current  stream-lines  are  deflected  toward  one  edge  of  the 
sheet.  This  is  called  the  Hall  Effect. 

(Ganot's  Physics,  art.  900.  Also  Campbell,  Modern 
Electrical  Theory,  pp.  76-78.) 

ELECTRO-OPTICAL  EFFECT  IN  DIELECTRICS. 

When  certain  dielectrics  are  subjected  to  electric  strain 
they  become  doubly  refracting.  Kerr  states  the  law  as 
follows : 

"  The  strength  of  the  electro-optical  action  of  a  given 
dielectric,  that  is  the  difference  in  the  path  of  the  ordinary 
and  extraordinary  rays,  for  unit  thickness  of  the  dielectric, 
varies  directly  as  the  square  of  the  resultant  electric  force." 

(Ganot's  Physics,  art,  997,) 


160  LAWS  OF  PHYSICAL  SCIENCE 

LAW  OF  SARASIN  AND  DE  LA  RIVE,  OF  «  MULTIPLE  RESONANCE." 

The  distance  between  two  nodes  on  a  resonator  changes 
in  changing  the  resonator  but  not  in  changing  the  oscillator. 
This  distance  (the  internode)  is  the  half  wave-length  of 
the  free  oscillations  of  the  resonator  only.  (These  results 
are  connected  with  the  rapid  damping  of  the  waves  which 
usually  occurs  on  the  oscillator.) 

(Vreeland,  Maxwell's  Theory  of  Wireless  Telegraphy, 
p.  62.) 

LAW    OF   DISTRIBUTION    OF    ELECTRO-MAGNETIC    RADIATION,    (i) 

At  any  point  at  a  distance  from  the  center  of  disturbance 
there  is  an  electric  and  magnetic  disturbance  at  right  angles 
to  the  line  drawn  from  the  center  of  disturbance.  The  elec- 
tric force  is  on  a  tangent  to  a  great  circle  of  a  sphere  with 
the  oscillator  at  its  center  and  which  has  its  poles  at  the 
intercepts  of  the  axis  of  the  oscillator  produced:  and  the 
magnetic  disturbances  lie  on  tangents  to  circles  parallel 
to  a  plane  perpendicular  to  the  oscillator. 

(Vreeland,  Maxwell's  Theory  of  Wireless  Telegraphy, 
pp.  76,  77.) 

LAW    OF    DISTRIBUTION    OF    ELECTRO-MAGNETIC    RADIATION.    (2) 

The  two  vibrations  (electric  and  magnetic)  are  trans- 
verse, as  in  light,  and  perpendicular  to  the  direction  of 
propagation  of  the  wave-front.  The  amplitude  of  these 
vibrations  varies  inversely  as  the  distance  and  their  in- 
tensity varies  inversely  as  the  square  of  the  distance  from 
the  oscillator.  The  vibrations  maintain  a  constant  direction 
as  do  those  of  polarized  light. 

(Vreeland,  Maxwell's  Theory  of  Wireless  Telegraphy, 
pp.  76,  77,) 


ELECTRICITY  AND  MAGNETISM  161 

RELATION  OF  MAGNETIC  FORCE   AND  A  MOVING  FARADAY  TUBE. 

A  Faraday  tube  in  motion  perpendicular  to  its  direction 
always  gives  rise  to  a  magnetic  force,  the  direction  of  motion, 
the  Faraday  tube  and  the  magnetic  force  being  mutually  at 
right  angles,  when  the  medium  is  isotropic. 

(Thomson,  .Elements  of  Electricity  and  Magnetism,  pp. 
479,  490-492.) 

POYNTING'S  LAW. 

When  a  conductor  carrying  a  current  is  in  an  electro- 
static field  the  transfer  of  energy  takes  place  through  the 
dielectric  along  paths  which  are  the  intersections  of  the 
equipotential  surfaces  of  the  electrostatic  field  with  the 
equipotential  surfaces  of  the  electromagnetic  field  due  to 
the  current. 

(Gray,  A  Treatise  on  Magnetism  and  Electricity,  p. 
421.) 

GENERALITY  OF  LAW  OF  INVERSE-SQUARES. 

It  may  be  stated  generally  that  the  intensity  of  an 
effect,  which  emanates  from  a  center  and  is  transmitted 
equally  in  all  directions,  is,  in  an  isotropic  medium,  inversely 
proportional  to  the  square  of  the  distance  from  the  source. 

(New  Century  Dictionary  under  word  LAW.) 


11 


VI 

LIGHT 


LIGHT. 

A  GENERALITY  IN  RADIATION. 

Every  fact  of  experience  and  every  consideration  of 
theory  go  to  prove  that  radiant  energy  of  all  wave-lengths, 
whether  called  electric,  heat-,  or  light-radiation,  travels  in 
space  free  from  ordinary  matter  with  the  same  velocity  and 
obeys  the  same  laws  of  propagation.  This  is  a  velocity  which 
would  be  determined  on  an  elastic  solid  theory  by  the  equation 

e 
df' 

where  e  is  the  elasticity  and  d  the  density  of  the  ether  of 
space. 

(Consult  Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8, 
pp.  2  and  94  et  seq.  Also  Preston,  The  Theory  of  Light, 
pp.  28-30.) 

LIGHT   DEFINED;   VELOCITY  OF. 

Light  consists  of  radiant  energy,  propagated  in  free 
space  with  the  velocity  common  to  all  radiant  energy  but 
having  wave-lengths  such  that  it  affects  the  human  eye. 

Calling  A  the  wave-length  and  N  the  frequency  of  the 
transverse  vibrations  the  velocity  is  very  approximately 
V==\N  =  3X1010  cms.  per  second  =  300,000  kilometers 
per  sec.  The  wave-length  of  visible  radiant  energy  lies 
within  the  limits  A  =  0.76/x,  and  A,  =  0.4/A,  and  the  frequency 
lies  in  the  limits  N  =  4  XlO14  and  N  =  7.5  X  101*.  (/*  = 
0.001  millimeter.) 

(Consult  Chwolson,  Trait  e  de  Physique,  Vol.  II,  Part  8, 
p.  29.  For  methods  of  determining  the  velocity  of  light, 
see  Preston,  The  Theory  of  Light,  Chap.  XIX,  p,  489  et  seq.) 

165 


166  LAWS  OF  PHYSICAL  SCIENCE 

RECTILINEAR  PROPAGATION  OF  LIGHT. 

Light  travels  in  straight  lines  through  a  homogeneous 
medium  if  the  rays  are  not  compelled  to  pass  through  any 
very  small  openings.  When  it  passes  through  media  of 
different  kinds,  it  does  not,  in  general,  travel  in  the  same 
straight  line  through  them  all. 

(Crew,  General  Physics,  p.  428.  Also  Ganot's  Physics, 
art.  513.) 

INTENSITY  OF  RADIATION. 

The  quantity  of  energy  which  traverses  in  the  unit  of 
time  unit  surface  normal  to  the  ray  is  here  chosen  to  define 
the  intensity  of  radiation.  This  intensity  is  inversely  pro- 
portional to  the  square  of  the  distance  to  a  point-source  of 
the  radiation.  The  intensity  of  radiation  received  on  an 
oblique  surface  is  proportional  to  the  cosine  of  the  angle 
which  the  ray  makes  with  the  normal  to  the  illuminated 
surface. 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  pp. 
24,  25.  Also  Ganot's  Physics,  art.  519.) 

LAWS  OF  REFLECTION. 

1.  The  angle  of  reflection  is  equal  to  the  angle  of  inci- 
dence. 

2.  The  incident  and  the  reflected  ray  are  both  in  the 
same  plane  which  is  perpendicular  to  the  reflecting  surface. 

(Ganot's  Physics,  art.  522.  Also  Preston,  The  Theory 
of  Light,  p.  74.  For  extensive  treatment  consult  Chwolson, 
Traite  de  Physique,  Vol.  I,  Part  2,  p.  168  and  Vol.  II, 
Part  8,  p.  112  et  seq.) 

REFLECTION  FROM  A  PLANE-MIRROR. 

"  The  image  of  a  point  in  a  plane-mirror  lies  on  the 
perpendicular  let  fall  from  the  point  to  the  mirror,  and 
lies  as  far  behind  the  mirror  as  the  point  lies  in  front  of 
the  mirror." 


LIGHT  167 

(Crew,  General  Physics,  p.  441.  Also  Ganot's  Physics, 
art.  524.) 

REAL  AND  VIRTUAL  IMAGES. 

Real  images  are  those  formed  by  the  reflected  rays  them- 
selves, and  virtual  images  are  those  formed  by  their  pro- 
longations. Eeal  images  can  be  received  on  a  screen,  virtual 
images  cannot.  Plane-mirrors  give  rise  to  virtual  images 
only. 

(Ganot's  physics,  art.  525.) 

LIGHT   REFLECTED   FROM  A   ROTATING   MIRROR. 

If  a  ray  of  light  falls  on  a  plane  mirror  and  the  mirror 
is  rotated  through  any  angle  about  an  axis  which  lies  in 
the  plane  of  the  mirror,  the  reflected  ray  is  rotated  through 
twice  the  angle. 

(Kimball,  College  Physics,  p.  563.  Also  Ganot's  Physics, 
art.  530.) 

LAW  OF  FERMAT,  OR  PRINCIPLE  OF  LEAST  TIME. 

When  light  passes  from  any  point  P  to  another  point 
P'  by  reflection  at  a  point  p  on  a  surface,  the  path  P  p  P' 
is  that  which  will  be  traversed  by  the  ray  in  the  least  time 
in  passing  from  P  to  P'  by  reflection  at  the  surface.  A 
similar  law  applies  for  the  paths  of  refracted  rays,  so  that 
when  light  travels  from  one  point  to  another  the  ray  pur- 
sues that  path  which  requires  the  least  time. 

(Preston,  The  Theory  of  Light,  p.  95.  Also  Chwolson, 
Traite  de  Physique,  Vol.  II,  Part  8,  pp.  114,  115.) 


168  LAWS  OF  PHYSICAL  SCIENCE 

REFLECTIONS   FROM   PORTIONS   OF   THE   SURFACE   OF   A  SPHERE. 

The  formulas  for  calculating  the  location  of  images 
formed  by  spherical  mirrors  are  simple  only  when  the 
spherical  surface  is  but  a  small  portion  of  the  surface  of 
a  sphere.  In  this  case  the  distance  s  from  a  point-source 
to  a  point  on  the  surface  of  the  mirror,  the  distance  s'  from 
the  image  to  the  same  point  on  the  surface,  and  the  radius 
r  of  the  spherical  mirror  bear  the  relations, 

s       s-r          11       2 
F  =  7^7"  or  T  +  7  =  T* 

o  £  —  o  DD  JT 

These  general  relations  hold  for  both  concave  and  con- 
vex surfaces  which  are  small  portions  of  a  sphere,  if,  calling 
the  center  of  the  mirror-surface  the  origin,  distances  to  the 
right  are  reckoned  positive  and  to  the  left  negative. 

(Chwolson,  Trait  e  de  Physique,  Vol.  II,  Part  8,  pp.  115- 
122.  Also  Crew,  General  Physics,  pp.  445,  446.) 

RELATIVE   SIZE    OF   OBJECT   AND   IMAGE. 

For  a  spherical  mirror, 

the  linear  dimensions  of  the  image 
the  linear  dimensions  of  the  object 

the  distance  from  a  point  on  the  mirror  to  the  image 
the  distance  from  the  same  point  on  the  mirror  to  the  object 

This  relation  holds  whether  the  image  be  real  or  virtual  and 
for  convex  as  well  as  concave  mirrors. 
(Ganot's  Physics,  arf.  540.) 

THE   CAUSTIC. 

When  the  aperture  of  a  spherical  mirror  much  exceeds 
10  degrees  the  rays  from  a  point-source  reflected  by  the 
mirror  suffer  spherical  aberration  by  reflection.  Every 
reflected  ray  cuts  the  one  adjacent  to  it  and  their  points 
of  intersection  form  in  space  a  curved  surface  which  is 
called  the  caustic  ~by  reflection. 

(Ganot's  Physics,  art.  542.) 


LIGHT  169 

REFLECTION   FROM   A  PARABOLIC   MIRROR. 

In  reflection  from  a  parabolic  mirror  all  rays  parallel 
to  its  axis,  after  reflection,  meet  at  the  focus  of  the  mirror ; 
and  conversely,  when  a  point-source  of  light  is  placed  at  the 
focus,  the  rays  incident  on  the  mirror  are  reflected  exactly 
parallel  to  the  axis,  and  their  intensity  tends  to  remain 
constant  at  all  distances. 

(Ganot's  Physics,  art.  544.) 

REFLECTION  FROM  MAT  SURFACES:  LAMBERT'S  LAW. 

Surfaces  from  which  the  diffusion  of  light  is  uniform 
are  said  to  be  mat. 

The  reflection,  from  such  surfaces,  of  light  incident 
under  a  given  angle  follows  the  law  of  cosines,  often  called 
"  Lambert's  law."  According  to  this  law  the  intensity 
of  the  light  diffusely  reflected  from  a  mat  surface  is  pro- 
portional to  the  cosine  of  the  angle  between  the  direction 
of  the  diffused  rays  under  consideration  and  the  normal  to 
the  surface.  However,  if  the  angle  of  incidence  be  varied, 
the  light  reflected  at  a  constant  angle  to  the  normal  is  not 
proportional  to  the  angle  of  incidence. 

(See  New  Century  Dictionary,  under  title  COSINE  LAW, 
and  the  sub-head,  Lambert's  Law  of  Cosines.) 

SELECTIVE  REFLECTION. 

Selective  reflection  is  reflection  in  which  the  incident  and 
reflected  rays  differ  in  composition.  Many  otherwise  trans- 
parent bodies  have  absorption-bands  in  their  spectra,  and 
waves  of  light  having  a  wave-length  to  which  the  body  is 
opaque  are  more  completely  reflected  than  the  others. 

(New  Century  Dictionary,  under  word  REFLECTION. 
Consult  Wood,  Physical  Optics,  pp.  352,  353.) 


170  LAWS  OF  PHYSICAL  SCIENCE 

REFLECTION  FROM  AN  ELEMENT  OF  A  NON-SPHERICAL  SURFACE. 

A  bundle  of  rays,  incident  normally  on  an  infinitely  small 
area  of  a  non-spherical  surface,  give  in  reflection  two  in- 
finitely small  right-angled  rectilinear  focal  lines,  parallel 
to  the  elements  of  lines  of  curvature  of  this  surface.  A 
bundle  of  such  rays  is  called  astigmatic. 

(Chwolson,  Trait  e  de  Physique,  Vol.  II,  Part  8,  pp.  125, 
126.  See  also  Preston,  The  Theory  of  Light,  p.  109.) 

VELOCITY   OF    LIGHT    IN    ORDINARY    MATTER. 

Light  travels  more  slowly  in  any  kind  of  ordinary  rratter 
than  in  vacuum.  Thus  for  red  rays, 

velocity  in  air 

— ; — T~. — -  =  1.329  (about) 

velocity  in  water 

,  velocity  in  air 

and  — , — rr^     — — — -- — .  ...    =  1.612  (about) 
velocity  m  carbon  bisulphide 

(Ganot's  Physics,  art.  561.  Also  Ames,  Theory  of 
Physics,  pp.  424-426.) 

LAWS   OF  REFRACTION:  SNELL'S   LAW. 

"  The  incident  and  refracted  rays  are  in  the  same  plane 
with  the  normal  to  the  surface ;  they  lie  on  opposite  sides 
of  it,  and  the  sines  of  their  inclinations  to  it  bear  a  con- 
stant ratio  to  one  another." 

Denoting  the  angles  of  incidence  and  refraction  by 
i  and  r  respectively,  the  relation  between  them  is  given 
by  the  formula, 

sin  i 

— —  =  f*. 
smr 

The  constant  ratio  ft  is  called  the  index  of  refraction. 
(Preston,  The  Theory  of  Light,  p.  88.) 

RELATION  OF  VELOCITIES  AND  INDEX  OF  REFRACTION. 

For  any  particular  medium  the  index  of  absolute  refrac- 
tion fji  varies  inversely  as  the  velocity  of  light  in  that 
medium. 


LIGHT  171 

Denoting  by  i  and  r  the  angles  of  incidence  and  refrac- 
tion respectively,  and  by  v  and  v'  the  velocities  in  ether 
and  the  medium  respectively, 

sin  i  ___v  _ 
sin  r  ~  v'~  M' 

(Preston,  The  Theory  of  Light,  pp.  89,  90.) 

RELATIONS  OF  VELOCITIES  IN  MEDIA  OTHER  THAN  ETHER. 

If  the  velocities  in  two  media  are  v±  and  v2  while  the 

V  V 

velocity  in  ether  is  v,  then    AH  =  •—  and  &  =  -  -,  or, 


Also  f4  sin  i  =  fi2  sin  r. 

For  any  number  of  media  the  continued  product  of  the 
relative  refractive-indices  of  n  substances  is  equal  to  the 
ratio  of  the  absolute  refractive-index  of  the  nth  substance 
to  that  of  the  first.  Or  in  a  formula, 

"U'-"*  .....  *Wn  =  77 

(Preston,  The  Theory  of  Light,  pp.  90,  91.  Also  Ames, 
Theory  of  Physics,  p.  425.) 

ATMOSPHERIC  REFRACTION. 

The  refractive  index  of  the  earth's  atmosphere  decreases 
as  we  ascend  and  for  this  reason  all  light  which  reaches 
us  from  stars  not  in  the  zenith  travels  in  curved  paths. 
The  effect  of  this  refraction  is  to  apparently  raise  the  stars 
toward  the  zenith.  The  mirage  results  from  the  double  view 
of  an  object  given  by  rays  reaching  the  eye  by  two  paths, 
one  nearly  direct  and  one  concave  upward,  giving  the  effect 
of  reflection. 

(Ganot's  Physics,  art.  551.  Also  Wood,  Physical  Optics, 
pp.  69,  70.) 


172  LAWS  OF  PHYSICAL  SCIENCE 

TOTAL  REFLECTION  AND  CRITICAL  ANGLE. 

Let  fji±  and  n2  be  the  indices  of  refraction  of  two  media, 
fjLt  being  greater  than  /x2.  Then,  if  a  ray  of  light  in  passing 
through  the  medium  of  greater  refraction  comes  to  the 
boundary  of  the  two  media,  it  can  only  pass  out  into  the 
medium  of  less  refraction  when  the  sine  of  the  angle  of 
incidence  (angle  between  ray  and  normal  to  surface  of 

separation  of  the  two  media)   is  less  than  — . 

f*i 
For  all  values  of  the  angle  of  incidence  greater  than 

this  the  ray  is  totally  reflected  from  the  plane  of  separation 
of  the  two  media. 

The  maximum  value  of  the  angle  of  incidence  which 
does  not  give  total  reflection  is  known  as  the  "  critical  " 
angle. 

(Ames,  Theory  of  Physics,  p.  427.  Also  Ganot's  Phys- 
ics, art.  550.) 

REFRACTION  AT  A  SINGLE  SPHERICAL  SURFACE. 

When  light  passes  from  a  point-source  in  a  medium  of 
refractive  index  /^  into  a  medium  of  refractive  index  p2 
bounded  by  a  spherical  surface,  then, 


where  r  —  radius  of  curvature  of  refracting  surface  —  r 
being  taken  positive  when  convex  side  of  surface  is  toward 
incident  ray  —  s  =  distance  from  apex  of  surface  to  point- 
source  —  s  being  taken  negative  when  on  same  side  of  sur- 
face as  point-source  —  and  s'  =  distance  from  apex  to  image 
—  where  s'  is  taken  negative  or  positive  according  as  it  is 
on  the  same  or  opposite  side  of  the  surface  as  the  point- 
source. 

(Crew,  General  Physics,  p.  458.) 


LIGHT  178 

POSITION  OF  IMAGE  FORMED  BY  REFRACTION  AT  A 
SPHERICAL    SURFACE 

The  position  of  the  image  when  rays  pass  from  air  into 
a  substance  bounded  by  a  spherical  surface  is  given  by  the 
formula, 

_e._JL  --"-1 

s'       s          r~' 

where  /x  =  index  of  refraction  of  substance  used,  s'  =  dis- 
tance from  apex  of  surface  to  image,  s  =  distance  from  apex 
of  surface  to  source  and  r  =  radius  of  curvature  of  spheri- 
cal surface. 

(This  formula  comes  from  the  one  just  preceding  by 
placing  P!  =  1  and  /*2  =  /x.) 

(Crew,  General  Physics,  p.  460.  Also  Ganot's  Physics, 
art.  552.) 

GENERAL  EQUATION  FOR  THIN  LENSES. 

— ,— —  =  (n-1)  (  —  -  — )    is  given  in  Ganot's  Physics, 
P      P  \ r     s  / 

art.  563,  as  a  general  formula  applicable  to  all  cases  both  of 
convex  and  concave  lenses. 

The  point  where  the  refracted  ray  cuts  the  axis  when 
the  incident  ray  is  parallel  to  the  axis  is  called  the  principal 
focus.  Its  distance  from  the  lens  is  the  focal  length  of  the 

lens.     The  power  of  a  lens  equals     ..    ,      T1 — — r.         The 

its  focal  length 

unit  of  power  is  the  ' i  dioptric  ' '  which  equals  the  power  of 
a  lens  of  focal  length  of  one  meter. 

(See  above  reference  for  interpretation  of  this  formula, 
or  consult  Crew,  General  Physics,  p.  466.) 


174  LAWS  OF  PHYSICAL  SCIENCE 

CHROMATIC    ABERRATION. 

The  index  of  refraction  for  a  given  substance  is  not  the 
same  for  all  wave-lengths,  or  colors  of  light.  Thus, 

M  Red  light,       Yellow  light,     Green  light, 

for,  C.  D.  F. 

Flint  glass  1.630  1.635  1.848 

Crown  glass  1.527  1.530  1.536 

This  failure  of  a  lens  to  bring  all  colors  to  the  same  focus 
is  known  as  ' '  Chromatic  Aberration. ' ' 

(Crew,  General  Physics,  p.  470.  Also  Ganot's  Physics, 
art.  594.) 

DEVIATION  OF  A  RAY  IN  PASSING  THROUGH  A  PRISM. 

The  angle  of  deviation  8  of  a  ray  passing  through  a 
prism  is  the  angle  between  the  incident  and  emergent  rays. 

8=(i1.-r1)  +  (i2-r2). 

Here  it  =  angle  of  incidence  (angle  between  normal  to 
first  face  of  prism  and  incident  ray), 

i2  =  angle  of  emergence  (angle  between  normal  to  second 
face  of  prism  and  emergent  ray), 

YI  =  angle  of  refraction  at  first  face  and 

r2  =  angle  of  refraction  at  second  face.  Also  angle  of 
prism  is,  a  =  r±  +  r2. 

The  angle  of  deviation  8  is  a  minimum  when  the  in- 
cident ray  makes  the  same  angle  with  the  first  face  of  the 
prism  as  the  emergent  ray  does  with  the  second  face  of 
the  prism. 

(Crew,  General  Physics,  p.  471.) 

VELOCITY   OF    LIGHT   IN    MEDIA    DEPENDS    UPON    WAVE-LENGTH. 

The  velocity  of  light  is  less  the  more  highly  refracting 
the  medium,  and  as  light  of  short  wave-lengths  is  more  re- 
fracted than  light  of  long  wave-lengths,  the  velocity  of  the 
former  is  less  than  that  of  the  latter  in  all  media  of  greater 
refractive  index  than  unity. 


LIGHT  175 

(Consult  Preston,  The  Theory  of  Light,  Chap.  XIX, 

pp.  513,  516.    Also  Wood,  Physical  Optics,  pp.  15,  16.) 

• 

DISPERSIVE  POWER. 

The  transformation  of  a  beam  of  parallel  rays  of  white 
light  into  a  divergent  pencil  of  light  of  different  colors  is 
knows  as  dispersion. 

The  dispersive  power  of  a  prism  is  the  ratio  of  the  angle 
of  separation  produced  of  two  selected  rays  to  the  mean 
deviation  (angle  between  incident  and  emergent  rays)  of 
the  two  rays, 


P 


dh-dA  _  nh-na 
d        "    n-1 


Here  nh  is  the  refractive  index  of  the  material  of  a  prism 
or  lens  for  violet  rays,  na  the  refractive  index  for  red  rays, 
n  the  refractive  index  for  mean  rays,  and  db,da,  and  d  are 
the  corresponding  deviations. 
(Ganot's  Physics,  art.  577.) 

VARIATION   OF   REFRACTIVE   INDEX   WITH    DENSITY    (GLADSTONE 
AND  DALE'S  LAW). 

When  a  substance  is  compressed  or  its  temperature 
varied  the  density  changes.  This  is  accompanied  by  a  cor- 
responding variation  in  the  refractive  index  such  that, 

refractive  index  —  1 

j r- =  a  constant 

density 

(Preston,  The  Theory  of  Light,  p.  131  et  seq.) 


i?6  LAWS  OF  PHYSICAL  SCIENCE 

BRIOT'S   FORMULA. 

Briot  has  determined  the  form  of  the  function  which 
expresses  the  variation  of  the  index  of  refraction  /u,  with 
wave-length  A.  His  formula  is, 


........ 

where  A,  B,  C,  etc.,  are  constants  depending  on  the  nature 
of  the  medium  and  diminishing  rapidly  as  we  proceed  to 
higher  terms,  and  K  is  another  constant. 

(Preston,  The  Theory  of  Light,  pp.  487,  488.) 

DOUBLE    REFRACTION:    OPTIC    AXES. 

In  all  transparent  crystals,  of  which  the  fundamental 
form  is  not  a  cube,  a  black  dot  seen  through  the  crystal 
appears  double  for  most  positions  of  the  crystal.  There 
are,  however,  in  some  crystals  one  and  in  others  two  direc- 
tions, along  which  the  dot  being  viewed  appears  single. 
This  direction  or  directions  constitute  the  optical  axis  or 
axes  of  the  crystal. 

This  phenomenon,  called  double  refraction,  is  very 
marked  in  the  uniaxial  crystal,  Iceland  spar  (calcium 
carbonate.) 

(Preston,  The  Theory  of  Light,  p.  299.  Also  Ganot's 
Physics,  art.  654.) 

DOUBLE     REFRACTION     IN    AN     UNIAXIAL     CRYSTAL:      GENERAL 
STATEMENT. 

Whatever  be  the  plane  of  incidence,  the  ordinary  ray 
always  obeys  the  two  general  laws  of  single  refraction. 

In  every  section  perpendicular  to  the  optic  axis,  the 
extraordinary  ray  follows  the  laws  of  single  refraction. 

In  every  principal  section  the  extraordinary  ray  follows 
the  second  law  of  refraction  only,  but  the  ratio  of  the  sines 
of  the  angles  of  incidence  and  refraction  is  not  constant. 

(Ganot's  Physics,  art.  656.) 


LIGHT  177 

DOUBLE    REFRACTION    IN    BIAXIAL    CRYSTALS:     GENERAL 

STATEMENT. 

When  a  ray  of  light  enters  a  biaxial  crystal  and  traverses 
it  in  any  direction  not  coinciding  with  an  optic  axis,  it 
bifurcates  and  generally  both  rays  are  extraordinary  rays. 
However,  in  a  section  of  the  crystal  at  right  angles  to  the 
medial  line  one  ray  follows  the  laws  of  ordinary  refraction, 
and  in  a  section  at  right  angles  to  the  supplementary  line 
the  other  ray  follows  the  laws  of  ordinary  refraction. 

(Ganot's  Physics,  art.  658.) 

CONICAL    REFRACTION. 

This  is  of  two  kinds,  Internal  conical  refraction  and 
external  conical  refraction.  For  the  former,  an  unpolarized 
ray  entering  a  crystal  along  an  axis  of  single  wave-velocity 
diverges  within  the  crystal  as  a  cone  and  emerges  as  a 
hollow  cylinder  of  rays.  These  can  be  received  on  a  screen 
as  a  ring  of  light  of  constant  diameter.  In  the  latter  case, 
if  a  conical  pencil  of  rays  enters  a  crystal  with  the  apex  of 
the  cone  on  the  face  of  the  crystal  and  with  its  axis  chosen 
parallel  to  the  axis  of  single  wave-velocity  the  emergent 
pencil  of  rays  (superfluous  rays  being  screened  off)  forms 
a  hollow  cone.  This  gives  on  a  screen  a  ring  of  light  which 
increases  in  diameter  with  the  distance  of  the  screen  from 
the  crystal. 

(Wood,  Physical  Optics,  pp.  257-259.) 

POLARIZATION  EFFECTED  BY  A  CRYSTAL. 

In  traversing  an  uniaxial  crystal  the  ordinary  ray  is 
polarized  in  the  plane  of  the  principal  section,  which  in- 
cludes this  ray  and  the  optic  axis.  The  extraordinary  ray, 
is  on  the  contrary,  polarized  in  a  plane  perpendicular  to  the 
plane  of  the  principal  section  passing  through  the  ray  and 
the  optic  axis.  The  polarization  is  complete  for  both  ordi- 
nary and  extraordinary  rays. 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  pp.  760, 
761.) 

12 


178  LAWS  OF  PHYSICAL  SCIENCE 

POLARIZATION  BY  REFLECTION. 

When  a  ray  of  ordinary  light  falls  at  an  angle  upon  a 
reflecting  surface  of  a  transparent  substance  the  reflected 
ray  is  more  or  less  polarized.  For  a  particular  angle  of 
incidence  the  polarization  of  the  reflected  ray  is  most  com- 
plete and  this  angle  is  called  the  polarizing  angle  for  the 
substance. 

(Ganot's  Physics,  art.  667,  or  Wood,  Physical  Optics, 
Chap,  IX.  See  p.  231.) 

ANGLE    OF    POLARIZATION:     BREWSTER'S    LAW. 

In  polarization  by  reflection,  the  polarizing  angle  of  a 
substance  is  that  angle  of  incidence  for  which  the  reflected 
polarized-ray  is  at  right  angles  to  the  refracted  ray. 

Brewster 's  law  states  that :  The  tangent  of  the  polarizing 
angle  for  a  substance  is  equal  to  the  index  of  refraction  of 
that  substance.  This  law  is  expressed  by  the  formula, 

sin  i 

H  =  tan  i  = -• 

cos  i 

As   In^  =  **  (Snell'S  law)j 

where  i  is  the  angle  of  incidence  of  the  reflected  ray  and  r 
the  angle  which  the  refracted  ray  makes  with  the  normal 
to  the  surface  of  separation  of  the  two  media,  it  follows 
that  i  -f-  r  =  90  °  ;  namely,  the  reflected  and  refracted  rays 
form  a  right  angle. 

(Ganot's  Physics,  art.  668.  Also  Wood,  Physical  Optics, 
p.  231.  Also  Preston,  The  Theory  of  Light,  pp.  303,  304.) 

LAW   OF   MALUS. 

When  a  pencil  of  light,  polarized  by  reflection  at  one 
plane-surface,  is  allowed  to  fall  upon  a  second  plane-surface 
at  the  polarizing  angle,  the  intensity  of  the  twice-reflected 
beam  varies  as  the  square  of  the  cosine  of  the  angle  between 
the  two  planes  of  reflection. 


LIGHT  179 

(Preston,  The  Theory  of  Light,  p.  305.  Also  Wood, 
Physical  Optics,  p.  235  et  seq.) 

POLARIZATION    OF    REFRACTED    LIGHT. 

The  relation  between  the  polarized  light  in  the  refracted 
pencil  and  that  in  the  reflected  beam  was  discovered  by 
Arago  and  is  stated  thus :  i '  When  an  unpolarized  ray  is 
partly  reflected  at,  and  partly  transmitted  through,  a  trans- 
parent surface,  the  reflected  and  transmitted  portions  con- 
tain equal  quantities  of  polarized  light,  and  the  planes  of 
polarization  are  at  right  angles  to  each  other." 

(Preston,  The  Theory  of  Light,  p.  304.) 

LAWS   OF  INTERFERENCE   OF  POLARIZED  LIGHT. 

"  1.  Two  rays  of  light  polarized  at  right  angles  do  not  inter- 
fere destructively  under  the  same  circumstances  as 
two  rays  of  ordinary  light. 

2.  Two  rays  polarized  in  the  same  plane  interfere  like 

two  rays  of  ordinary  light. 

3.  Two  rays  polarized  at  right  angles  may  be  brought  to 

the  same  plane  of  polarization  without  thereby 
acquiring  the  quality  of  being  able  to  interfere  with 
each  other. 

4.  Two  rays  polarized  at  right  angles,  and  afterwards 

brought  to  the  same  plane  of  polarization,  interfere 
like  ordinary  light  if  they  originally  belonged  to  the 
same  beam  of  polarized  light. " 
(Preston,  The  Theory  of  Light,  p,  308.) 


180  LAWS  OF  PHYSICAL  SCIENCE 

THE   PLANE   OF  POLARIZATION. 

' '  ,The  plane  of  polarization  is  defined  as  the  particular 
plane  of  incidence  in  which  the  polarized  light  is  most 
copiously  reflected." 

When  a  ray  of  light  is  incident  at  the  angle  of  polariza- 
tion on  one  mirror  and  the  reflected  ray  is  received  on  a 
second  mirror  at  the  polarizing  angle,  the  ray  is  most 
copiously  reflected  from  the  second  mirror  when  the  two 
mirrors  are  parallel.  In  this  case  the  plane  of  reflection 
coincides  with  the  plane  of  polarization.  /  The  vibrations 
of  plane-polarized  light  are,  however,  in  a  direction  at 
right  angles  to  the  plane  defined  as  the  plane  of  polarization. 

(Wood,  Physical  Optics,  p.  233.) 

ELLIPTICAL     POLARIZATION. 

Light  may  be  plane,  circularly  or  elliptically  polarized. 
Plane  and  circular  polarization  may  be  treated  as  special 
cases  of  elliptical  polarization.  /  Circular  polarization  results 
from  the  simultaneous  presence  at  a  point  of  two  rectangu- 
lar vibrations  of  the  same  period  but  differing  in  phase 
by  a  quarter  period.  In  elliptical  polarization  both  the 
phase  and  amplitude  of  the  vibrations  differ. 

(Wood,  Physical  Optics,  Chap,  XI,  p.  266  et  seq.  Also 
Preston,  The  Theory  of  Light,  pp.  417,  418.) 

ROTATION   OF  PLANE    OF  POLARIZATION. 

Rotation  of  the  plane  of  polarization  occurs  when  plane- 
polarized  light  is  transmitted  through  quartz  in  the  direc- 
tion of  its  optic  axis,  and  this  property  is  also  possessed 
by  many  other  substances,  including  many  liquids  and 
vapors.  Some  rotate  the  plane  to  the  right  (looking  along 
the  direction  of  propagation  of  the  light)  and  are  called 
"  Dextrogyrate,"  some  to  the  left  and  are  called  "  Levo- 
gyrate." 

(Preston,  The  Theory  of  Light,  p.  425.  Also  Ganot's 
Physics,  art.  687.) 


LIGHT  181 

BIOT'S    LAWS. 

The  amount  of  rotation  of  plane-polarized  light  passed 
through  a  quartz  crystal  or  other  rotating  substance  is 
proportional  to  the  thickness  traversed  by  the  ray.  The 
rotation  effected  by  two  plates  is  the  algebraic  sum  of  the 
rotations  produced  by  each  separately.  The  rotation  aug- 
ments with  the  refrangibility  of  the  light,  and  is  approxi- 
mately proportional  to  the  inverse  square  of  the  wave- 
length. 

(Preston,  The  Theory  of  Light,  p.  426.  Also  Wood, 
Physical  Optics,  pp.  384,  385.) 

POLARIZATION    BY    EMISSION    AND    DIFFUSION. 

Rays  emitted  at  an  oblique  angle  from  incandescent 
platinum  are  partially  polarized  perpendicularly  to  the 
plane  of  emission.  Rays  falling  on  a  surface  which  is  not 
absolutely  mat,  in  their  diffuse  reflection  are  partially 
polarized. 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p.  740.) 

RELATION  OF  INTENSITY  AND  AMPLITUDE. 

If  the  periods  of  two  vibrations  are  the  same,  then  the 
intensities  of  the  rays  are  in  the  ratio  of  the  squares  of 
the  amplitudes  of  the  vibrations.  Thus, 


where  J,  J\  are  intensities  and  a,  ax  are  amplitudes. 

(On  the  electromagnetic  theory  of  light,  amplitude  of 
vibration  would  correspond  to  the  maximum  potential- 
difference  between  opposite  ends  of.  the  oscillator  giving 
rise  to  the  radiant  energy.) 

(Consult,  Preston,  The  Theory  of  Light,  pp.  43,  44.) 


182  LAWS  OF  PHYSICAL  SCIENCE 

A  NATURAL   RAY   REPLACED   BY   TWO   RAYS   POLARIZED   AT 
RIGHT  ANGLES. 

Two  rays  rectilinearly  polarized,  which  replace  a  natu- 
ral ray,  have  amplitudes  continually  variable  and,  in  gen- 
eral, unequal  at  each  instant ;  the  mean  value  of  the  square 
of  the  amplitude  determines  the  quantity  of  radiant  energy 
(light  intensity)  in  the  two  rays. 

Thus,  A2  =  %J,  where  J  is  the  intensity  of  the  natural 
radiation  and  A2  is  the  mean  value  of  the  square  of  the 
amplitude  of  the  component  rays. 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p,  696.) 

LAW  OF  LAMBERT  FOR  EMISSION  FROM  A  SURFACE  OF  RADIANT 

ENERGY. 

The  quantity  of  radiant  energy  emitted  in  the  unit  of 
time,  by  an  element  of  the  surface  of  a  body,  in  any  given 
direction,  is  proportional  to  the  cosine  of  the  angle  between 
this  direction  and  the  normal  to  the  surface  of  the  radiating 
body.  Thus,  J<?  =  J  cosp  where  J  is  the  total  quantity 
of  energy  emitted  normally  and  J<p  the  quantity  emitted 
in  the  direction  making  the  angle  $  with  the  normal. 
(It  is  because  of  this  law  that  a  sphere  heated  to  be  uni- 
formly luminous  appears  equally  bright  at  the  central  point 
of  its  disc  and  at  its  boundary.) 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p.  36.) 

LAW    OF    ABSORPTION    OF    LIGHT. 

When  a  ray  enters  a  homogeneous  medium  the  quantity 
of  light  of  a  given  wave-length  which  is  absorbed  is  pro- 
portional to  the  thickness  of  the  medium  traversed,  and 
the  amount  of  light  which  passes  through  a  number  of 
equal  layers,  diminishes  in  geometrical  progression  as  the 
number  of  layers  increases  in  arithmetical  progression. 

(Consult  Preston,  The  Theory  of  Light,  Chap.  XVIII. 
See  pp.  469-471.  Also  Wood,  Physical  Optics,  Chap.  XIV. 
See  pp.  350,  351.) 


LIGHT  183 

DOPPLER-FIZEAU    PRINCIPLE. 

If  a  source  of  light  and  an  observer  approach,  the  fre- 
quency of  the  disturbance,  as  it  passes  the  observer,  is 
increased  and  the  wave-length  diminished;  if  they  recede 
the  reverse  is  true.  On  approach  the  spectrum  lines  are 
shifted  toward  the  violet  and  on  recession  they  are  shifted 
toward  the  red. 

(Wood,  Physical  Optics,  Chap.  I.    See  p.  19.) 

HUYGENS'    PRINCIPLE. 

The  wave-front  in  a  train  of  light-waves  is  a  surface  of 
disturbance  which  results  from  and  envelops  (i.e.,  is 
tangent  to)  the  secondary  waves  sent  out  by  each  particle 
lying  in  the  wave-front  at  an  earlier  instant. 

(Wood,  Physical  Optics,  Chap.  II,  p.  21  et  seq.  Also 
see  Ames,  Theory  of  Physics,  pp.  401-403.  Also  Preston, 
The  Theory  of  Light,  p.  60.) 

HUYGENS'    PRINCIPLE    OF    SUPERPOSITION. 

Any  number  of  separate  disturbances  (light- waves)  may 
be  propagated  through  one  another  in  the  same  portion  of 
the  medium.  Each  emerges  from  that  portion  as  if  it  had 
not  been  encountered  by  others.  Rays  of  light  from  all 
objects  round  about  cross  each  other's  paths  in  all  sorts 
of  ways,  but  each  travels  on  as  if  the  others  did  not  exist. 

(Preston,  The  Theory  of  Light,  p.  45.) 

STOKES'  LAW. 

The  effect  or  intensity  of  an  elementary  wave  at  an  ex- 
ternal point  varies  as  (1  +  cos  0),  where  0  is  the  obliquity, 
or  angle  between  the  wave-normal  and  the  line  joining  the 
point  to  the  center  of  the  elementary  wave.  Thus  the  effect 
only  vanishes  for  0  =  TT,  that  is,  for  points  directly  behind 
the  wave. 

(Preston,  The  Theory  of  Light,  p.  62.) 


184  LAWS  OF  PHYSICAL  SCIENCE 

PRESSURE  OF  RADIANT  ENERGY;  GENERAL  CASE. 

When  radiant  energy  (of  any  wave-length)  falls  nor- 
mally on  a  perfectly  black  surface  the  pressure  exerted  on 
unit  area  is  numerically  equal  to  the  total  quantity  of  radi- 
ant energy  contained  in  the  unit  of  volume.  If  the  surface 
is  perfectly  reflecting  the  pressure  is  twice  as  great.  If  the 
radiation  falls  on  the  surface  making  an  angle  ^  with 
the  normal  to  the  surface  then  the  pressure,  per  unit  area  is, 

p  =  e  (1  +  a)  cos2  ??, 

E 

where       e  =  y-       is  the  energy  in  the  unit  of  volume  and  a 

is  the  fraction  of  the  energy  reflected,    a  =  1  for  a  perfectly 
reflecting  and  a  =  o  for  a  perfectly  black  surface. 

(Chwolson,  Traite  de  Physique,  Vol.  II,  Part  8,  p.  84.) 

LAW     OF     KIRCHHOFF-CLAUSIUS     ON     THE     RELATION     BETWEEN 
EMISSIVE    POWER   AND    THE    MEDIUM. 

The  emissive  power  of  perfectly  black  bodies  is  propor- 
tional to  the  square  of  the  index  of  refraction  /*  of  the  sur- 
rounding medium. 

Thus,  e  =  /*2E, 

where  E  is  the  emissive  power  of  a  perfectly  black  body  in 
vacuum  and  e  its  value  in  any  medium. 

(Chwolson,  Trait  e  de  Physique,  Vol.  II,  Part  8,  p.  83.) 

DIFFRACTION    OF    LIGHT. 

In  passing  through  a  very  narrow  aperture  in  a  screen 
a  ray  of  light  spreads  out  on  either  side  of  the  line  of 
rectilinear  propagation.  This  phenomenon  is  known  as 
diffraction 

It  is  also  observed  under  other  circumstances,  as  when 
light  passes  the  edge  of  a  body,  in  which  case  luminous  rays 
are  bent  into  the  shadow. 

(Ganot's  Physics,  arts.  660,  661.  For  a  detailed  treat- 
ment, see  Wood,  Physical  Optics,  Chap.  VII,  pp.  150-211.) 


LIGHT  185 

INTERFERENCE  OF  LIGHT-RAYS. 

When  light  passes  through  two  small  openings  in  a 
screen  which  are  adjacent  and  the  two  transmitted  rays 
meet  each  other  at  any  point  under  a  small  angle  the  two 
trains  of  waves  either  annul  or  strengthen  each  other, 
giving  dark  and  light  bands  on  a  screen.  In  the  former 
case  the  distances  from  any  point  to  the  two  openings 
differ  by  a  half  wave-length  or  a  multiple  of  this,  and  in 
the  latter  case  by  a  whole  wave-length  or  a  multiple.  This 
reciprocal  action  of  two  wave-trains  is  called  interference. 

Michelson's  Interferometer  is  an  instrument  for  the 
measurement  of  lengths  by  means  of  the  phenomena  result- 
ing from  the  interference  of  two  rays  of  light.  The  instru- 
ment permits  the  introduction  of  any  relative  retardation 
between  interfering-pencils  of  light  and  allows  observation 
to  be  made  of  interference  bands  corresponding  to  a  large 
difference  of  path. 

(Ganot's  Physics,  art.  659.  Also  Preston,  The  Theory 
of  Light,  p.  202.  Wood,  Physical  Optics,  Chap.  VIII,  pp. 
212-229.  Also  Ames,  Theory  of  Physics,  pp.  395-397.) 

INTERFERENCE  IN  THIN  FILMS. 

If  two  plates  of  glass  are  put  together  so  as  to  form  a 
thin  wedge  of  air,  destructive  interference  of  any  mono- 
chromatic light  will  take  place  where  the  thickness  of.  the 
wedge  equals  an  even  number  of  quarter  wave-lengths  and 
reenforcement  occurs  where  this  thickness  equals  an  odd 
number  of  quarter  wave-lengths. 

(Crew,  General  Physics,  pp.  502-505.  For  detailed 
account  of  these  phenomena,  see  Wood,  Physical  Optics, 
Chap,  VI,  p.  100  et  seq.) 


186  LAWS  OF  PHYSICAL  SCIENCE 

ANOMALOUS    DISPERSION. 

In  transparent  substances  in  which  the  dispersion  is 
normal,  the  refractive  index  increases  as  the  wave-length 
decreases,  but  in  substances  which  show  selective  absorp- 
tion, the  refractive  index  for  short  waves  on  the  blue  side 
of  an  absorption  band  is  often  less  than  the  index  for  red 
light  on  the  other  side  of  the  band.  This  phenomenon  has 
been  named  "  anomalous  dispersion. M 

(Wood,  Physical  Optics,  Chap.  V.    See  p.  95.) 

KUNDT'S  LAW  IN  ANOMALOUS  DISPERSION. 

This  law  states  that  on  approaching  an  absorption  band 
from  the  red  side  of  the  spectrum  the  refractive  index  is 
abnormally  increased  by  the  presence  of  the  band,  while 
if  the  approach  is  from  the  blue  side  the  index  is  abnormally 
decreased. 

(Wood,  Physical  Optics,  Chap.  V.    See  p.  96.) 

FRESNEL'S   LAW. 

"  In  crystals  the  velocities  of  the  two  light- waves  are 
proportional  to  the  largest  and  smallest  radii  vectors  of 
the  oval  section  of  the  wave-surface,  made  by  a  plane 
through  the  center  of  the  surface  and  parallel  to  the  wave- 
front.  " 

(New  Century  Dictionary,  under  word  LAW.  For  a 
detailed  treatment  of  wave-velocity  in  crystals,  consult 
Wood,  Physical  Optics,  Chap.  X.  See  pp.  246-257.) 

NON-REVERSIBLE    VISION. 

If  two  Nicols  are  mounted  with  their  directions  of  vibra- 
tions at  an  angle  of  45°  and  between  them  is  a  medium  in 
a  magnetic  field  of  such  strength  that  the  plane  of  rota- 
tion is  turned  45°,  light  will  be  stopped  by  the  second 
Nicol  in  going  one  way  and  be  wholly  transmitted  in  going 


LIGHT  187 

the  reverse  way.  It  is  thus  possible  to  produce  an  arrange- 
ment whereby  we  can  see  without  being  seen  when  mono- 
chromatic light  is  used. 

(Wood,  Physical  Optics,  p.  401.) 

NEWTON'S  RINGS. 

When  a  lens  of  large  radius  of  curvature  R  is  pressed 
upon  a  plane-plate  of  plate-glass  and  viewed  by  reflected 
monochromatic  light,  alternate  dark  and  bright  rings  are 
produced.  If  t  is  the  thickness  of  air-film  between  the 
lens  and  plate  and  r  is  the  radius  of  a  ring,  then, 

_J*_ 
~2R' 

With  sunlight  the  rings  are  many  colored. 

(Wood,  Physical  Optics,  Chap.  VI.  See  p.  131.  Also 
Ganot's  Physics,  art.  664.) 

THE  ZEEMAN^EFFECT. 

When  a  source  of  monochromatic  radiation  is  placed  in 
a  strong  magnetic  field  the  lines  of  the  spectrum  are  widened 
and  many  spectral  lines  are  broken  up,  in  the  Zeeman- 
effect,  into  multiple  lines. 

(For  description  and  explanation  of  the  "Zeeman- 
effect  "  see  Wood,  Physical  Optics,  pp.  403-410.) 

STOKES'    LAW    REGARDING    FLUORESCENCE. 

Stokes'  law  asserts  that  the  wave-length  of  the  light 
emitted  by  a  fluorescent  body  always  exceeds  that  of  the 
exciting  light. 

(Wood,  Physical  Optics,  Chap.  XVIII.    See  p.  434.) 


188  LAWS  OF  PHYSICAL  SCIENCE 

DRAPER'S  LAW  OF  VISIBILITY. 

Draper  considered  that  all  bodies  begin  to  become  visible 
by  self-emission  of  light  at  the  same  temperature.  He 
stated  that  this  temperature  is  525°  C. ;  but  the  beginning 
of  visibility  is  somewhat  dependent  upon  the  condition  of 
the  eye.  The  first  visual  sensation  received  from  heated 
bodies  is  now  known  to  be  a  sense  of  brightness  to  which, 
definite  color  cannot  be  assigned.  The  first  color-sensation 
received  coincides  with  the  region  in  the  spectrum  of 
maximum  luminosity — a  yellow  green. 

Lummer  and  others  have  shown  that  Draper's  law  does 
not  hold  true. 

(New  Century  Dictionary  under  word  LAW.  See  the 
interesting  account  of  the  supposed  law  of  Draper  in  Chwol- 
son,  Traite  de  Physique,  Vol.  II,  Part  8,  pp.  33,  34  and 
80,  81.) 

LE  CHATELIER'S  LAW  OF  RADIATION. 

The  empirical  law  of  Le  Chatelier  for  the  intensity  of 
radiation  of  red  light  is  represented  by  the  formula, 

3210 

I  =  106-7.  T~^> 

where  T  is  the  absolute  temperature  of  the  radiating  body 
and  I  is  the  intensity. 

(Consult  Burgess  and  Le  Chatelier,  Measurement  of 
High  Temperatures,  p.  302.) 

TALBOT'S   LAW. 

Helmholtz  states  this  law  thus:  If  any  part  of  the 
retina  is  excited  with  intermittent  light,  recurring  periodi- 
cally and  regularly  in  the  same  way,  and  if  the  period  is 
sufficiently  short,  a  continuous  impression  will  result,  which 
is  the  same  as  that  which  would  result  if  the  total  light 
received  during  each  period  were  uniformly  distributed 
throughout  the  whole  period. 

(Bulletin  of  the  Bureau  of  Standards,  Vol.  II,  p.  1.) 


LIGHT  189 

ASTRONOMICAL    ABERRATION. 

The  phenomenon  known  as  the  astronomical  aberration 
of  light  is  the  apparent  displacement  of  a  star  due  to  the 
resultant  effect  of  the  velocity  of  light  and  the  motion  of 
the  earth. 

This  phenomenon  was  discovered  and  an  explanation 
of  it  given  by  Bradley  in  1728.  The  maximum  displacement 
of  a  star  from  this  cause  is  20.51"  of  arc. 

(Preston,  The  Theory  of  Light,  pp.  12  and  518-520. 
See  treatment  of  this  problem  in  Lorentz,  The  Theory  of 
Electrons,  art.  155.  For  value  of  the  "  Constant  of  Aber- 
ration," see  Smithsonian  Physical  Tables,  p.  109.) 

THE    ZONE-PLATE. 

The  zone-plate  is  a  flat  surface  which  will  bring  light, 
transmitted  through  it,  to  a  focus  in  the  manner  of  a  convex 
lens.  It  is  made  by  describing  on  a  glass  plate  a  system  of 
circles  having  a  common  center  with  radii  which  increase 
proportionally  to  the  square  roots  of  the  natural  numbers, 
and  blackening  all  odd  numbered  rings.  The  smaller  the 
zones  the  shorter  is  the  focal  length. 

(See  illustration  and  description  of  this  interesting 
optical  device  in  Wood,  Physical  Optics,  p.  31.) 

THE    NICOL'S    PRISM. 

The  Nicol  's  prism  is  a  most  valuable  device  for  studying 
polarized  light.  It  is  constructed  from  a  rhombohedron  of 
Iceland  spar  which  has  been  cut  in  two  along  a  parallel 
plane  and  the  two  pieces  rejoined  in  their  original  position 
with  a  layer  of  Canada  balsam  or  air  between.  It  transmits 
the  extraordinary  ray  only  and  polarizes  light  completely. 

(Ganot's  Physics,  art.  674.) 


BIBLIOGRAPHY  AND  INDEX 


BIBLIOGRAPHY 

American  Journal  of  Science. 

Ames,  J.  S.,  Theory  of  Physics,  1896. 

Appell,  Traite  de  Mecanique  Rationnelle,  1902-1904. 

Bedell  and  Crehore,  Alternating  Currents,  1893. 

Bulletins  of  the  Bureau  of  Standards. 

Burgess  and  Le  Chatelier,  Measurement  of  High  Temperature,  1912, 

3rd  Ed. 

Campbell,  Norman  R.,   Modern  Electrical  Theory,   1913,  2nd  Ed. 
Christie,  C.  V.,  Electrical  Engineering,  1913. 
Chwolson,  Trait^  de  Physique,  1906-1914,  five  volumes. 
Clausius,  R.,  On  Heat,  edited  by  Hirst,  1867. 
Crew,  Henry,  General  Physics,  1909. 
Edser,  Heat,  1908. 
Encyclopedia  Britannica,  10th  Ed. 
Fourier,  Joseph,  The  Analytical  Theory  of  Heat.    Trans,  by  Freeman, 

1878. 

Ganot's  Physics,  1905,  17th  Ed. 
General  Electric  Review. 

Gray,  Andrew,  A  Treatise  on  Magnetism  and  Electricity,  1898. 
Gulliver,  G.  H.,  Metallic  Alloys,  1913,  2nd  Ed. 
Helmholtz,  Popular  Scientific  Lectures.    Sensations  of  Tone. 
Hering,  C.,  Conversion  Tables,  1904. 
Houstoun,  An  Introduction  to  Mathematical  Physics. 
Jeans,  J.  H.,  Electricity  and  Magnetism,  1908.    The  Dynamical  Theory 

of  Gases,  1904.    Theoretical  Mechanics,  1907. 
Journal  of  The  Franklin  Institute. 

Kaye  and  Laby,  Physical  and  Chemical  Constants,  1911. 
Kimball,  A.  L.,  College  Physics,  1912. 
Lagrange,  Mecanique  Analytique. 
Laplace,  P.  S.,  Traite  de  Mecanique  Celeste. 
Lommel,  Experimental  Physics,  1899. 
Lord  Kelvin,  Mathematical  and  Physical  Papers. 
Lorentz,  H.  A.,  The  Theory  of  Electrons,  1909. 
Mach,  E.,  Science  of  Mechanics,  1902. 
Maxwell,  James  Cle,rk,  Theory  of  Heat,  1904.     Treatise  on  Electricity 

and  Magnetism,  1881,  2nd  Ed. 

193 


194  BIBLIOGRAPHY 

Merriman,  M.,  Treatise  on  Hydraulics,  1903. 

Nernst,  Walter,  Theoretical  Chemistry,  1911. 

"New  Century  Dictionary. 

Northrup,  E.  F.,  Methods  of  Measuring  Electrical  Resistance,  1912'. 

Planck,  Vorlesungen  uber  die  Theorie  Der  Warmestrahlung. 

Poynting  and   Thomson,   Heat,   1904.     Properties   of  Matter,    1902. 
Sound. 

Preston,  Thomas,  Theory  of  Heat,  1904. 

Rankine,  W.  J.  M.,  The  Steam  Engine,  1882. 

Richardson,  O.  W.,  The  Electron  Theory  of  Matter,  1912. 

Roscoe  and  Schorlemmer,  Treatise  on  Chemistry,  Vol.  I,  1911;  Vol. 
II,  1907. 

Routh,  Elementary  Rigid  Dynamics,  1905,  7th  Ed. 

Scientific  Memoirs,  edited  by  J.  S.  Ames. 

Smithsonian  Physical  Tables,  1916,  6th  Revised  Ed. 

Spencer,  L.  J.,  The  World's  Minerals,  1911. 

Steinmetz,  C.  P.,  Alternating  Current  Phenomena,  1900. 

Stokes,  G.  G.,  Mathematical  and  Physical  Papers. 

Thomson,  J.  J.,  Elements  of  Electricity  and  Magnetism,  1904.     Cont 
duction  of  Electricity  Through  Gases,  1906,  2nd  Ed. 

Thomson  and  Tait,  Treatise  on  Natural  Philosophy,  Part  I,  1886; 
Part  II,  1883. 

Transactions  American  Electrochemical  Society. 

Transactions  Connecticut  Academy. 

Vreeland,  Maxwell's  Theory  and  Wireless  Telegraph,  1904. 

Walker,  James,  Introduction  to  Physical  Chemistry,  1913. 
Washburn,  Edward  W.,  An  Introduction  to  the  Principles  of  Physi- 
cal Chemistry,  1915. 

Watson,  A  Text-look  of  Physics,  1899,  1st  Ed. 
Webster,  The  Dynamics  of  Particles  and  of  Rigid,  Elastic  and  Fluid 

Bodies,  1904,  2nd  Ed. 
Wood,  R,  W,,  Physical  Optics,  1905,  1st  Ed. 


INDEX 

(Numbers  refer  to  pages) 


Aberration,  chromatic,  174 

of  light,  189 
Absolute  scale  of  temperature,  63 

zero,  definition  of,  63 
Absorption,  electric,  116 
of  gases  by  liquids,  83 

by  solids,  84 
of  light,  182 
Absorptive  power,  104 
Acceleration,  centripetal,  9 

of    chemical    reactions    with 

temperature,  100 

Accelerations,  compounded  by  par- 
allelogram rule,  8 
Acid  and  base,  neutralization,  96 
Acids,  relative  avidity  of,  96 
Acoustic  attraction  and  repulsion, 

53 

Action,  principle  of  least,  24 
Addition    of    simple    sound-vibra- 
tions, 56 

Additive  property  of  dilute  solu- 
tions, 94 
Adiabatic  expansion,  74 

relations,  75 
Alternating  current  power,  139 

currents,  Kirchhoff's  laws 

applied  to,  138 
Ohm's  law  applied  to,  139 
Ampere's    law    for    the    magnetic 

field,  140 

Amplitude,     necessary     to     make 
Bound-waves  audible,  54 


Analogies,  electric   and  magnetic, 
145 

Analogue   between   osmotic  press- 
ure and  gas-pressure,  92 

Analogues  in  translation  and  rota- 
tion, 12 

Anomalous  dispersion,  186 

Kundt's  law  on,  186 

Archimedes'  principle,  31 

Areas,    conservation    of,    D'Arcy's 

statement,  23 
Mach's  statement,  23 

Astigmatic  rays,  170 

Astronomical  aberration,  189 

Atmospheric  refraction,   171 

Attraction  and  repulsion,  acoustic, 
53 

Audibility,  limits  of,  54 

Avogadro,  his  gas-law,  76 

Avidity  of  acids,  96 

Bells,  vibration  of,  53 
Bernoulli's  theorem,  34 
Bertrand's  principle  of  similitude, 

17 

Berzelius,  gave  name  catalysis,  102 
Black-body,  definition  of,  104 
Blagden's    law    on    depression    of 

freezing  point,  99 
Boiling,  86 

and  volatilization,  86 
point,  dissolved  salts  raise,  98 
Boltzmann-Maxwell,    on    equipar- 

tition  of  energy,  76 

195 


196 


INDEX 


Boyle's  law,  for  gases,  72 

variations  from,  73 

Boys,  C.  V.,  his  value  of  Newton- 
ian constant,  4 

Bradley,  on  aberration,  189 

Brewster's  law,  regarding  polariz- 
ing angle,  178 

Biot's  laws,  on  rotation  of  plane- 
polarization,   181 

Briot's  formula,  176 

Callendar,  his  formula,  65 
Capacities,  parallel  and  series  com- 
binations of,  121 
Capacity  and  conductance,  general 

relation  between,  136 
and  resistance,  a  relation  be- 
tween, 136 
Capillary  action,  Jurin's  law,  37 

law  of,  38 

corrections    of    mercury    col- 
umns, 38 

Carnot's  theorem,  64 
Catalysis,  102 
Caustic,  the,  168 

Cavendish,   measurement  of  New- 
tonian constant,  4 
Centripetal  acceleration,  9 
Charged     bodies,     force     between 

varied  by  medium,  116 
Charges,    decay   of   in   dielectrics, 

152 

Charles'  law,  for  gases,  74 
Chemical  action,  balanced,  90 

decomposition,  progress  of,  90 
combination  of  elements,  96 
reactions,  acceleration  of  with 

temperature,  100 
Chromatic  aberration,  174 


Circuits,  kinetic  energy  of  two,  150 

mutual  action  between,  148 
Clapeyron,  his  constant  and1  work 

of  expansion  of  a  gas,  80 
his  equation  for  gas-constant, 

79 

Clausius,  on  entropy-increase,  71 
Coefficients  of  induction,  148 
Coil  to  give  maximum  self-induc- 
tion, 149 

Coils,  resistance-values  from,  129 
Colloids  and  crystalloids,  102 
Cooling,   Dulong   and  Petit's   con- 
clusions on,  63 
Newton's  law  of,  62 
Combinational  tones,  54 
Composition      by      parallelogram 

rule,  8 

Compound  pendulum,  law  of,  13 
Condensers,   combined  ia  parallel 

and  series,  121 
Condensation  on  nuclei  of  vapor, 

88 
Conductance  and  capacity,  general 

relation  between,  136 
Conduction,       electrification      by, 

113 
Conductor,  force  at  surface  of  a 

charged,  119 

magnetic  force  within  a,  141 
-resistance  changes  with  spe- 
cific resistance,  130 
Conductors  and  dielectrics,  in  un- 

uniform  field,  120 
and  insulators,  114 
four- terminal,  131 
interaction  of  electric,  152' 
Conductivity,  Onnes  on,  133 

Ostwald's    law   of   molecular, 
95 


INDEX 


197 


Conservation    of    areas,    D'Arcy's 

statement,  23 
Mach's  statement,  23 
of  energy,  principle  of  the,  28 
of  living  forces,  22 
of  matter  or  mass,  21 
of  moment  of  momentum,  22 
of  momentum,  21,  22 
of  movement  of  center  of  grav- 
ity, 20 
Conservative    system,    work    done 

by,  20 
systems,   19 
Consonance  and  dissonance,  Helm- 

holtz  on,  57 
Constant  heat-summation,  law  of, 

101 

Newtonian,  4 
Constraint,   comment  on  principle 

of  least,  27 

Gauss's  principle  of  least,  26 
Cosine,    law  of  the,   in   radiation, 

106 
Contact  difference  of  potential,  137 

electricity,  137 
Continuity,  principle  of,  35 
Conversion  of  energy  in  a  conduc- 
tor, 132 

Convertibility  of  energy,  70 
Cord,  transverse  vibrations  of,  51 
Corresponding  states,  78 
Coulomb's  law  of  electric  intensity, 

117 

Critical  temperature,  91 
Cryohydric  temperature,  99 
Crystalloids  and  colloids,  102 
Crystals,  law  of  interfacial  angles 

for,  110 

Crystal-zones,  Neumann's  law  of, 
110 


Crystal,  polarization  by,  177 

Curie's  law  of  magnetic  suscepti- 
bility, 91 

Currents,  law  of  resolution  of,  153 
mechanical  action  of,  152 
mutual  relations  of,  152 
thermoelectric,  154 

Cycloidal  pendulum,  14 

D'Alembert's    and    Gauss'    princi- 
ples compared,  27 
principle,  25,  26 

Dalton,    his    law   for    solution    of 
mixed  gases,  83 

Dal  ton's  laws  of  mixture  of  gases, 
76 

D'Arcy's  principle  regarding  areas, 
23 

Definite   resistance,   condition  for, 
130 

Definition,  of  a  quantity  of  heat,  61 
of  temperature,  61 

Definitions,  for  absolute  zero  and 

temperature,  63 
of  sound,  noise,  etc.,  45 

Delaroche  and  Berard,  on  molecu- 
lar heat,  77 

Deville,    his  use  of  word  "  disso- 
ciation," 90 

Dielectric     constant,     or     specific 
inductive  capacity,  120 

Dielectrics,  decay  of  charges  in,  152 

Diffraction,  of  light,  184 
of  sound,  50 

Diffusion  in  liquids,  92 

Dilute  solutions,  Kohlrausch  on,  94 

Diosmose,  92 

Dispersive  power,  175 

Displacement,  electric,  120 

Distribution,  law  of,  89 

Doppler-Fizeau  principle,  183 


198 


INDEX 


Doppler's     principle,     applied     to 

sound,  48 
Double  refraction,  176 

in  biaxial  crystals,  177 
in  uniaxial  crystals,  176 
Draper  on  visibility,  188 
Dulong  and  Petit,   on  velocity  of 

cooling,  63 
Dulong  and  Petit's  law  of  thermal 

capacity,  66 

Ear,  effect  of  sound-waves  on,  55 
Earnshaw's   theorem  on   stability, 

121 

Echoes,  49 
Efflux,  quantity  of,  33 

Torricelli's  theorem,  33 
Effusion  of  gases,  85 
Elastic   medium,    velocity   of    dis- 
turbance in,  17 
Elasticity,  coefficient  of,  15 
Elements,     chemical     combination 

of,  96 

periodic  system  of,  97 
Elliptical  polarization,  180 
Electric  absorption,  116 

charges,  law  of  repulsion  of, 

114 
circuit    and    magnetic     shell 

compared,  140 
and  magnet  compared,  142 
current,   action  of   a  magnet 

on,  140 

displacement,  120 
equilibrium,  116 
field,  effects  of  ununiform,  on 

conductors,  120 
energy  of,  120 

intensity  at  a  surface,  117 
inside  a  conductor,  117 


Electric  intensity,  inside  and  out- 
side conductors  and  dielec- 
trics, 121 

stress  in  a  medium,  119 
Electrical   conductor,   longitudinal 

motion  in,  143 
and  magnetic  analogies,  145 
Electricity,  compared  with  incom- 
pressible fluid,  118 
positive  and  negative,  114 
Electrification,  by  conduction,  113 
by  friction,  by  induction,  113 
Electrified    system,  work  done  in 

displacement  of,  118 
Electrodes,     Kirchhoff     on     inter- 
change of,  131 

Electrolytic  decomposition,  a  prin- 
ciple of,  128 
Electrolysis,  conservation  of  energy 

in,  127 

Faraday's  first  law  of,  128 
Faraday's  second  law  of,  128 
little  affected  by  pressure,  127 
Electromagnetic  radiation,   distri- 
bution of,  160 
Electromotive  force,  115 

acts    on    electricity    only, 

151 

and  current,  resolved,  138 
impressed  on  a  circuit,  151 
independent  of  nature  of 

conductor,  147 
-force  series,  126 
work  done  by,  139 
forces  in  series,  136 
Electron,  unit  of  negative  electric- 
ity, 114 
Electronic   emission,    Richardson's 

law  of,  156 

Electro-optical  effect  in  dielectrics, 
159 


INDEX 


199 


Emissive  power,  definition  of  abso- 
lute, 103 

Kirchhoff-Clausius  on,  184 
monochromatic,  103 
e/m,  value  of,  114 
Energy,  conservation  of  in  electro- 
lysis, 127 
conversion  of  mechanical  into 

heat  in  conductor,  132' 
convertilibility  of,  70 
equipartition  of,  76 
intrinsic,  70 
Kelvin's  theorem  of  minimum, 

24 
magnetic     and     electrokinetic 

compared,  150 
minimum  potential,  14 
of  a  system  of  conductors,  118 
of  electric  field,  120 
of  rotation,  10 
Poynting's  law  on  transfer  of, 

161 

pressure  of  radiant,  158 
principle  of  the  conservation 

of,  28 

transformation  of,  70 
Engineering    equation,     a    funda- 
mental, 154 
Entropy,  70 

Clausius  on  increase  of,  71 
Eotvos,  law  of,  91 
Equation,    fundamental    engineer- 
ing, 154 

of  continuity,  35 

Equations,  of  mechanics,  basic,  6 
Equilibrium,  condition  of  tempera- 
ture, 62 
for  a  system,  6 
for  liquids,  32 
electric,   116 


Equilibrium,     Gibbs*    criteria    of 

thermal,  99 
law  of  relative  proportions  in, 

89 

of  a  system,  D'Alembert's  con- 
dition for,  25,  26 
of  floating  bodies,  31 
of   liquids   in   communicating 

vessels,  32 
three  states  of,  28 
Equalization  of  temperature,  62 
Equipartition  of  energy,  76 
Equipotential    magnetic    surfaces, 

142 

Evaporation,  work  done  by,  84 
Exchanges,  Prevost's  theory  of,  109 
Expansion,  adiabatic,  74 

of  a  gas,  Clapeyron's  constant, 

80 

of  anisotropic  bodies,  65 
of  bodies  with  heat,  65 
of  liquids,  66 
Extension,  Young's  modulus,  19 

Falling  bodies,  law  of,  5 
Faraday,  the,  128 

on    impossibility    of    absolute 

charge,  116 

on  rotation  of  plane  of  polar- 
ized light,  158 
-tube,     relation    of    magnetic 

force  and  moving,  161 
Faraday's  first  law  of  electrolysis, 

128 

second  law  of  electrolysis,  128 
Fermat,  his  principle  of  least  time, 

167 
Floating  bodies,  action  of  surface 

tension  on,  41 
equilibrium  of,  31 


200 


INDEX 


Flow  of  heat,  for  steady  state,  66 
from  point-source,  68 
general  equation  for,  67 
in  crystalline  medium,  68 
of  water  in  pipes,  34 
through  capillary  tubes,  37 
Films,  light-interference  in,  185 
First  law  of  thermodynamics,  68 
Fluid,  incompressible,  and  electric- 
ity compared,  118 
resistance  to  motion  of  solid 

in,  35 

Fluorescence,  Stokes'  law  on,  187 
Force,  action  of  impulsive,  18 

between  charged  bodies  varied 

by  medium,   116 
between  magnetic  poles,  122 
-functions,  22 
on  magnetic  pole  exterior  to 

conductor,  140 

Forces,   compounded   by  parallelo- 
gram-rule, 8 
conservation  of  living,  22 
equilibrated      and      effective, 
D'Alembert's    principle    of, 

25,  26 
equilibrifum — condition        for 

three,  Lami's  theorem,  8 
of   cohesion  and   surface   ten- 
sion, 39 
vectorial  addition  of  magnetic 

and  electrostatic,  144 
Form  assumed  by  liquid  mass,  39 
Fourier,  on  compounding  harmonic 

motions,  54 

Four-terminal  conductors,   131 
Freezing    point    lowered,    Raoult's 

law,  98 

points,  depression  of  by  press- 
ure, 98 


Fresnel's  law,   186 

Friction,  electrification  by,  113 

sliding,  18 

rolling,  18 

statical  and  kinetic,  18 
Fusion,  of  solids  and  metals,  99 

Gas — constant,  the,  79 
for  perfect  gas,  79 
Gas-laws,  applied  to  solutions,  93 
Gas,  Joule's  law  regarding,  77 
internal  friction  of,  78 
-molecules,  velocity  of,  81 
number  of  molecules  in  a,  82 
pressure  and  energy  of,  80 
a  gram-molecule  of,  78 
specific  heat  of  given  volume 

of,  77 

-temperature-scale,  64 
work  of  expansion  of,  80 
Gases,  absorption  of  by  solids,  84 
basic     equation     and     kinetic 

theory  of,  80 

combinations  of,  by  volume,  97 
Dalton's  laws  of  mixture  of,  76 
diffusion  of,  Graham's  law,  85 
effusion  of,  85 

five  fundamental  laws  of,  72 
Graham's  law  of  diffusion  of, 

85 
Henry's  law  of  absorption  of, 

83 

occlusion  of,  85 
solution  in  liquid  of  mixed,  83 
specific  heat  of,  77 
work    performed    when    two, 

mix,  82 
Gauss'  and  D'Alembert's  principle 

compared,  27 
principle   of   least   constraint, 

26 
theorem,  117 


INDEX 


201 


Gay-Lussac   and   Humboldt's   law, 

97 

Gay-Lussac's  law,  for  gases,  74 
General  principle,  mechanical,  7 
Gibbs,  his  phase-rule,  100 

on  thermal  equilibrium,  99 
Graham,    named   crystalloids    and 

colloids,  102 
Graham's  law  of  diffusion  of  gases, 

85 

Gladstone  and  Dale's  law,  on  re- 
fractive index,  175 
Gravitation,  Newton's  law  of  uni- 
versal, 4 
Gravity,  acceleration  of,  4 

conservation   of   movement  of 

center  of,  20 
pressure      produced     by,      in 

liquids,   31 
principle  of  motion  of  center 

of,  21 

Guldberg  and  Waage,  on  mass-ac- 
tion, 88 
Gyration,  radius  of,  11 

Hall  effect,  159 
Hamilton's  principle,  27 
Harmonic  motion,  simple,  12' 
Heat,  absorption  of  radiant,   102 
and  energy,  difference  between 

absorbed,  70 

condition    for    minimum    pro- 
duction of,  132 
definition  of  quantity  of,  61 
flow,  general  equation  for,  67 
law  of,  for  steady  state,  66 
of,  from  point-source,  68 
of,  in  crystalline  medium, 

68 
Hess  on  disengagement  of,  90 


Heat,  in  a  conductor,  Joule's  law 

for,  132 

intensity  of  radiant,   106 
of  formation,  101 
produced  by  radium,  72 
radiation  at  oblique  angle,  106 

Heats  of  reaction,  101 

Helmholtz,  on  consonance  and  dis- 
sonance, 57 
on  Talbot's  law,  188 
results    of    his    researches    on 
sounds,  55 

Henry's  law,  of  gas-absorption,  83 

Hess,  on  disengagement  of  heat,  90 

Hooke's  law,  19 

Huygens'  principle,  183 

of  superposition,  183 

Hydrion,  formation  of,  95 

Hydrodynamical  theorem,  37 

Hydrostatic  paradox,  32 

Hysteresis,  magnetic,  126 

Image,  by  refraction  at  spherical 

surface,  173 

relative  size  of  object  and,  168 
Images,  real  and  virtual,  167 
Impact,  between  two  bodies,  15 
for  perfectly  elastic  bodies,  16 
Newton's  law  of,  15 
velocities  after  and  before,  16 

before  and  after,  15 
Impulse  of  a  force,  18 
Inclined  plane,  descent  on,  5 
Index  of  refraction,  170 
Induction,    coefficient    of    mutual, 

148 

coefficients,  148 

coil  to  give  maximum  self,  149 
electrification,     produced     by, 
113 


INDEX 


Induction,   law  of  magnetic,    145, 

146 

magnetic,  125 
Inertia,  moment  of,  10 

principal  axes  of,  10 
Insulators  and  conductors,  114 
Intensity  and  amplitude,  relation 

of,  181 

of  sound,  45,  46 
of  radiation,  166 

Interaction  of  currents  and  mag- 
nets, 144 

Interference  in  films,  185 
of  light-rays,  185 
of  polarized  light,  179 
of  sound,  50 
Internal  pressure  in  a  conductor, 

law  of,  142 
Intrinsic  energy,  total  not  known, 

70 
Inverse  square  law,  generality  of, 

161 

Ions  and  molecules,  similarity  in 
behavior  of,  94 

Joule's  equivalent  of  heat,  132 

law  for  heat  in  conductors,  132 
respecting  a  gas,  77 
of  capillary  action,  37 

Kater's  pendulum,  14 
Kepler's  first  law,  7 
second  law,  7 
third  law,  7 

Kelvin,  his  absolute  temperature- 
scale,  63 
his  minimum  energy- theorem, 

24 

Kerr  effect,  159 

Kinetic  energy  of  two  circuits,  150 
theory,  fundamental  equation 
in,  80 


Kirchhoff-Clausius,     on     emissive 

power,  184 

Kirchhoff's  black-body,  104 
law  of  radiation,  104 

propositions         deducible 

from,  105 

for  electric  currents,  138 
Steinmetz's   extension   of, 

138 
Theorem    on    interchange    of 

electrodes,  131 

Kohlrausch,    on    dilute    salt-solu- 
tions, 94 

Kundt's    law    of    anomalous    dis- 
persion, 186 

Lambert's  law  of  light — emission, 

182 
of  reflection,   169 

Lami's  theorem,   8 

Latent  heat  of  vaporization,  86 

Least  action,  principle  of,  24 
constraint,  comment  on,  27 
Gauss'  principle  of,  26 
time,  for  passage  of  light-ray, 
167 

LeChatelier's  law  of  radiation,  188 

Lenses,  formula  for  image  by,  173 

Lenz's  law,  146 

Lever,  law  of,  8 

Light,  absorption  of,  182 
defined,  165 
defr action  of,  184 
-rays,  interference  of,  185 
rectilinear  propagation  of,  166 
reflection    of,    from    rotating 

mirror,  167 

rotation  of  plane  of,  158 
velocity  of,  in  matter,  170 

Limits  of  audibility,  54 

Line-integrals,   fundamental,    144 


INDEX 


203 


Line    of    magnetic    induction    de- 
fined,  146 
Liquid,    boiling    point    raised    by 

salt,  98 

-mass,  form  assumed  by,  39 
rotation  by  optically  active,  97 
-surface,  distance  between  two 

elements  of,  40 
normal  pressure  on,  39 
Liquids,   condition  of   equilibrium 

of,  32 

diffusion  in,  92 
equilibrium    of,    in   communi- 
cating vessels,  32 
expansion  of,  66 
forces  of  cohesion  of,  33 
theorem  of  Bernoulli  regard- 
ing, 34 

vapor-pressure  of  mixed,  87 
Longitudinal  motion  in  an  electri- 
cal conductor,  143 
Lummer,  on  Draper's  law,  188 

Mach's  statement  of  conservation 

of  areas,  23 

'Magnet,  action  of  current  on,  140 
compared  with  small  electric 

circuit,  142 

magnetic  force  due  to  a,  124 
Magnetic   and   electric  analogues, 

126 

and  electro-kinetic  energy  com- 
pared,   150 

energy  is  potential  energy,  148 
field,  Ampere's  law  for,  140 

magnetism  induced  by,  124 
fields  due  to  a  sphere  and  mag- 
net compared,  124 
force  and  a  moving  Faraday 
tube,  161 


Magnetic    force,    derived    from    a 

potential,  140 
due  to  a  magnet,  124 
exterior  to  a  linear  con- 
ductor,  140 

within  a  conductor,  141 
hysteresis,  126 
induction,  125 

law  of,  145,  146 
line  of,  defined,  146 
pole    and    equipotential    sur- 
faces, 142 
work  done  by,  143 
poles,  law  of  force  between,  122 
quantities,    relation    between, 

125 
shell,   compared  with  electric 

circuit,  140 
susceptibility,  Curie's  law  on, 

91 
Magnetism,  five  definitions,  122' 

induced  by  magnetic  field,  124 
total  charge  of,  123 
Magnetized     sphere    and     magnet 

compared,  124 

Magnetomotive    force    and    induc- 
tion in  solenoid,  153 
Magnets  and  electric  currents,  in- 
teraction of,  144 
Magnus,  law  of,  157 
Malus,  law  of,  178 
(Mass-action,  law  of,  88 
Mass,  its  relation  to  weight,  4 
Mat  surfaces,  reflection  from,  169 
Matter  or  mass,  conservation  of,  21 
Maxwell,  law  of  molecular  veloci- 
ties, 81 

on  definition  of  line  of  mag- 
netic induction,  146 


204 


INDEX 


Maxwell,  on  K  =  /ia,  158 

on  internal  friction  of  a  gas, 

78 

Mechanical  action  of  currents,  gen- 
eral principle,  152 
force    acts    on    conductor    not 

current,  147 

at  surface  of  charged  con- 
ductor, 119 

parallelogram  representa- 
tion of,  147 
principle,  general,   7 
Mechanics,  basic  equations  of,  6 
Medium,    effect    of,    on   force   bet- 

tween  charged  bodies,  116 
Melting  point,  effect  of  pressure  on, 

98 

Mercury  columns,  capillary  correc- 
tions for,  38 

Mariotte's  law,  for  gases,  72 
Metals,  resistance-temperature  re- 
lations of,  134,  135 
Michelson's  interferometer,  185 
Millikan,  on  electronic  charge,  114 
Minimum  heat  condition,  132 

potential  energy,  14 
Mirage,  the,  171 

Mirror,  light  reflected  from  rotat- 
ing, 167 

reflection  from  parabolic,  169 
Molecular  heat,  Delaroche  and  Ber- 

ard  on,  77 
rotation,  97 
species,  distribution  among 

several,  89 
surface-energy,   91 
Molecules,  number  of  in  a  gas,  82' 
Moment  of  inertia,  10 
Momentum,    changed    by    an    im- 
pulsive force,  18 


Momentum,  conservation  of,  21,  22 

moment  of,  22 
Motion,    continuous,    produced   on 

magnets,  144 
of  center  of  gravity,  21 
Newton's  first  law  of,  3 
second  law  of,  3 
third  law  of,  3 
perpetual,  impossible,  2'0 
simple  harmonic,  12' 
vortex,  36 
Musical  pipe,  vibrations  produced 

by,  52 
scales,  principle  of,  56 

tones,  Helmholtz  on,  55 
Multiple  resonance,  160 
Mutuality  of  phases,  100 

Natural  ray  replaced  by  polarized 

ray,  182 
Neumann's    law   of   crystal   zones, 

110 
on  molecular  specific  heat, 

66 
Neutralization  of  acid  and  base,  96 

progress  of,  95 
Newton,  attraction  due  to  a  sphere, 

4 

experimental  law  of  impact,  15 
first  law  of  motion,  3 
his  rule  for  velocity  of  sound, 

48 

law  of  cooling,  62 
law  of  universal  gravitation,  4 
on  velocity  of  sound  in  gases,  48 
rings,  187 

second  law  of  motion,  3 
third  law  of  motion,  3 
velocity     of     disturbances     in 
elastic  medium,  17 


INDEX 


205 


Nicol's  prism,  189 
Nodes  and  loops  in  organ-pipes,  52 
Nuclei,    effect    of    on   vapor — con- 
densation, 88 

0«clusion  of  gases,  by  solids,  85 

Octave,  the,  55 

Ohm,  G.  S.,  law  of,  for  sound,  55 

Ohm's  law, 

respecting       sound-vibra- 
tions, 55 

Steinmetz's   extension   of, 
139 

Onnes,  on  superconductivity,  133 

Optic  axes,   176 

Optical    rotation,    Oudeman's    law 
of,  98 

Organ-pipes,  nodes  and  loops  in,  52 

Oscillation,  convertibility  of  points 
in  a  pendulum,  14 

Osmose,  92 

Osmotic  pressure  and  gas-pressure 
compared,  92 

Ostwald's  law  of  molecular  conduc- 
tivity, 95 

Oudeman's  law  of  optical  rotation, 
98 

Paradox,  hydrostatic,  32' 
Parrallelogram  rule,  8 
Partial  pressure,  law  of,  84 

of  vapor,  84 
Pascal's  law,  32 

Paschen's  law  for  sparking  poten- 
tial, 119 
Peltier  effect,  155 

measure  of,  156 
Pendulum,  convertibility  of  points 

of  suspension  of,  14 
cycloidal,  14 
law  of  compound,  13 


Pendulum,  simple,  13 
Periodic   system   of    chemical   ele- 
ments, 97 

Perpetual  motion  impossible,  20 
Phase,   change   of   in   sound-reflec- 
tion, 49 
rule,  100 
Phases,  law  of  the  mutuality  of, 

100 
Photo-chemical    reaction,    law    of, 

109 

Pipes,  flow  of  water  in,  34 
Plane,  descent  on  an  inclined,  5 
of  polarization,  180 
mirror,  reflection  from,  166 
Planetary   motion,   Kepler's    three 

laws  of  7 

Planck's  law  of  spectral  distribu- 
tion, 109 

Plates,  vibration  of,  52 
Platinum-resistance    thermometer, 

65 

Poiseuille's  law  of  flow,  37 
Polarization,  angle  of,  178 
by  a  crystal,  177 
by     emission     and     diffusion, 

181 

by  reflection,  178 
elliptical,  180 
galvanic,  127 
of  refracted  light,  179 
plane  of,  180 
rotation  of  plane  of,  180 
Polarized  light,   laws  of   interfer- 
ence of,  179 

rays   and   natural   rays   com- 
pared, 182 
Positive   and   negative    electricity, 

114 

Potential,  definition  and  meaning 
of,  115 


206 


INDEX 


Potential,  due  to  a  magnetic  sole- 
noid, 123 

due    to    a    system    of    point- 
charges,  116 

energy  changes,  118 
Power  in  a  circuit,  139 
Poynting's  law  on  transfer  of  en- 
ergy, 161 
Pressure  and  energy  of  a  gas,  80 

-difference    on    two    sides    of 
soap-film,  40 

law  of  partial,  84 

little  affects  electrolysis,   127 

of  gram- molecule  of  gas,  78 

of  radiant  energy,  107,  184 

of  radiation,  107 

of  sound,  53 

produced  in  liquid,  31 

Rankine  on  variation  of,  75 

within  an  electric  conductor, 

142 

Prevost's  theory  of  exchanges,  109 
Principal  axes  of  inertia,  10 

focus,  173 

Prism,  deviation  of  ray  by,  174 
Propagation  of  sound,  45 
Pure  tone,  54 

Quantities,  relation  between  mag- 
netic, 125 
Quantum    hypothesis,    109 

Radiant    energy,     Lambert's     law 

for,  182 
pressure  of,  general  case, 

184 
heat,  absorption  of,  102 

-intensity  of,  106 
Radiation,  distribution  of  electro- 
magnetic, 160 


Radiation,  intensity  of,  166 
Kirchhoff's  law  of,  104 
-law,  propositions  deducible 

from,  105 

LeChatelier'a  law  of,  188 
of  heat  at  oblique  angle,  106 
-pressure    of    radiant    energy, 

107 

Stefan-Boltzmann  law  of,  107 
velocity  of,  in  ether,  165 
Radium,  heat  produced  by,  72 
Radius  of  gyration,  11 
Rankine,  on  absorbed  heat  and  en- 
ergy, 70 

on   second   law  of   thermody- 
namics, 69 

on  variation  of  pressure,  75 
Raoult,    his    law    on    lowering    of 

vapor-pressure,  93 
Raoult's    law    on    depression    of 

freezing  point,  98 
Ratio  of  units  of  electricity,  157 
Rayleigh,   on  pressure  exerted  by 

sound-waves,  53 
reciprocation  theorem,  20 
Reaction,  heats  of,  101 
Reciprocation  theorem,  Rayleigh's, 

20 

Reflection  from  mat  surfaces,  169 
non-spherical  surface,  170 
parabolic  mirror,  169 
plane  mirror,  166 
spherical  surface,  168 
laws  of,  of  light,   166 
phase-change  in  sound-,  49 
light  polarized  by,  178 
selective,  169 

total  and  critical  angle  of,  172 
Refracted    light,    polarization    of, 
179 


INDEX 


307 


Refraction,   at  a  single   spherical 

surface,  172 
atmospheric,  171 
conical,  177 
double,  176 

double,  in  biaxial  crystals,  177 
in  uniaxial  crystals,  176 
image    by,    at    spherical    sur- 
face, 173 

index  of,  for  sound,  50 
laws  of,  170 
of  sound,  50 
Refractive  index,  variation  of  with 

density,  175 
Regnault,     constant     of     thermal 

capacity,  66 
his  data  for  specific  heat  of 

gases,  77 

his  data  on  gases,  74 
on  sound-intensity,  46 
Relative     proportions     in     equili- 
brium, 89 
Resistance   and    capacity,    general 

relation  between,  136 
and         specific         resistance, 

changes,  130 

combinations,     total     obtain- 
able, 130 

condition  for  a  definite,  130 
temperature-coefficient,       defi- 
nitions     and      applica- 
tions, 134 
relations  for  metals,  134, 

135 

to  motion  through  fluid,  35 
-values  obtainable  from  coils, 

129 
Resistances  in  parallel,  130 

in  series,  129 

Resonance,  principle  of  in  sound,  49 
Resultant    forces    of    cohesion    at 
surface  of  liquid,  33 


Richardson's     law     of     electronic 

emission,  156 
Richter's    law,   on   interchange   of 

constituents  of  salts,  97 
Ripples,  on  surface  of  liquids,  41 

speed  of,  42 

Rods  and  plates,  vibration  of,  52 
Rolling  friction,  18 
Rotation  of  plane  of  polarization, 

180,  158 

Sarasin  and  de  la  Rive  on  multi- 
ple resonance,  160 
Scale  of  gas-thermometer,  64 
Scales,  principle  of  musical,  56 
Screw  and  wrench,  principle  of,  9 
Second  law  of  thermodynamics,  69 
Selective  reflection,  169 
Similar  systems,  vibration  of,  52 
Similitude,     Bertrand's     principle 

of,  17 
Simple  harmonic  motion,  12 

pendulum,  13 

Size  of  object  and  image,  168 
Sliding  friction,  18 
SnelPs  law  of  refraction,  170 
Solenoid,  induction  in,  153 

potential  due  to,  123 
Soap-film,  formulae  relating  to,  40 
Solution  in  liquid  of  mixed  gases, 

83 

Solutions,    additive    property    of 
dilute,  94 

gas-laws  applied  to,  93 
Sound,  defraction  of,  50 

intensity  of,  45,  46 
of,  in  tubes,  46 

interference  of,  50 

pressure  of,  53 

propagation  of,  45 

refraction  of,  50 


208 


INDEX 


Sound,  resonance  in,  49 

transmission,   expansions   and 

contractions  in,  46 
velocity  of,  general  principle, 

47 

of,  in  air,  47 
of,  in  liquids  and  solids, 

48 

-waves,  amplitude  to  be  audi- 
ble, 54 
effect  on  ear  of  a  system 

of,  55 

reflection  of,  49 
Sparking  potential,  Paschen's  law 

for,  119 
Specific  heat  of  gases,  77 

of  a  given  volume  of  gas, 

77 

molecular,  66 
inductive  capacity,  120 
inductive   capacity  and   index 

of  refraction,  158 
Spectral  distribution,  Planck's  law 

of,  109 

Wien's  law  of,  108 
Sphere,   fall   of   small,   in   viscous 

medium,  25 

reflection  from  surface  of,  168 
Stability,  Earnshaw's  theorem  on, 

121 
States,  theorem  of  corresponding, 

78 
Steel  spheres,  time  of  contact  when 

impacting,  16 
Stefan-Boltzmann's  radiation  law, 

107 

Steinmetz,  extension  of  Kirchhoff's 
laws,  138 
of  Ohm's  law,  139 
his  law  for  hysteresis,  126 


Stokes'  law  for  fall  of  sphere,  25 
on  fluorescence,  187 
on  light  intensity,  183 
Stress  and  strain,  Hooke's  relation 

for,  19 

state  of  electric,  119 
Sum  of  partial  tones,  56 
Super-conductivity,  133 
Surface-tension,      action      of,      on 
floating  bodies,  41 
work  of  forces  of  cohesion 

in,  39 
System,  definition  of  conservative, 

19 

work  done  by  a  conservative, 
20 

Talbot's  law,  188 

Temperature,  absolute  scale  of,  63 

by  resistance- thermometer,  65 

by  gas-thermometer,  64 

critical,  91 

definition  of,  61 

effect  of,  on  balanced  chemical 
action,  90 

-equilibrium,  62 

equalization  of,  62 

of  fusion,  99 

the  cryohydric,  99 
Theorem  of  Carnot  on  efficiency,  64 
Thermal     capacity,     Dulong     and 

Petit's  law  of,  66 
Thermodynamic      temperature- 
scale,  63 
Thermodynamics,  first  law  of,   68 

second  law  of,  69 
Thermoelectric   law,   for   pairs   of 
metals,    155 

power  and  law  of,  154 

currents,  154 

inversion,  157 


INDEX 


209 


Thomson  effect,  156 

theorem  on  electrolysis,  127 

Time     of     contact     of     impacting 
spheres,  16 

Tone,  pure,  54 

Tones,  combinational,  54 
the  sum  of  partial,  56 

Torricelli's  theorem,  33 

Total  reflection,  172 

Transformation  of   energy,   Rank- 
ine's  statement,  70 

Translation     and     rotation,     ana- 
logues in,  12 

Transverse  vibrations  of  a  cord,  51 

Trouton's   law,  on   latent  heat   of 
vaporization,  86 

Tubes,  sound- intensity  in,  46 

v,  the  ratio  of  units  of  electricity, 

157 

Van  der  Waals'  equation,  general- 
ized, 78 
formula,  73 

combining     Boyle's     and 

Charles'  law,  74 
Vapor,  condensation  of  saturated, 

88 

-formation  in  vacuum,  87 
-pressure,    in    communicating 

vessels,  87 
of  mixed  liquids,  87 
statements   regarding,   88 
Raoult's  law  on  lowering 

of,  93 

Vaporization,  latent  heat  of,  86 
Vector  relations,  148 
Velocities  and  relation  of  index  of 

refraction,  170 

Velocities,   compounded  by   paral- 
lelogram rule,  8 
in  media  other  than  ether,  171 


Velocities,   Maxwell's  law  of  mo- 
lecular, 81 
of  gas-molecules,  81 
principle  of  virtual,  24 
Velocity  of  disturbance  in  elastic 
medium,   17 
of  light,  165 

dependence   of,    on   wave- 
length,   174 
in  matter,  170 
of  mass-action,  88 
of  sound  and  air-density,  47 
in  air,  47 

general  principle,  47 
1  in  liquids  and  solids,  48 

Newton  on,  48 
of  transverse  wave  along  cord, 

51 
Vectorial     addition    of     magnetic 

forces,  144 
Vector-potential,  150 
Vena  contracta,  33 
Vertical  distance  between  elements 

of  liquid-surface,  40 
Vibration  of  bells,  53 
of  plates,  53 
of  rods  and  plates,  52 
Vibrations,     addition     of     simple 

sound,  56 
number   produced  by  musical 

pipe,  52 
of  a  cord,  51 
of  similar  systems,  52 
Vinal  and  Bates,  their  value  for 

the  "  Faraday,"  128 
Vires  vivae,  22 

Virtual  velocities,  principle  of,  24 
Visibility,  Draper  on,  188 
Vision,  non-reversible,  186 
Volatilization,  86 


210 


INDEX 


Volta,  on  con  tact- difference  of  po- 
tential, 137 
Vortex  motion,  36 

Water-waves,  speed  of,  41 
Wave-length  of  maximum  energy, 

Wien'a  law  of,  108 
Wave,  velocity  of  transverse,  along 

cord,  51 

Waves,  speed  of  water,  41 
Weight  and  mass,  proportionality 

between,  4 
loss  of,  by  immersion  in  fluid, 

31 

Wenner,  on  four-terminal  conduc- 
tors, 131 

Wiedemann-Franz  ratio,  133 
Wien's     displacement     law,     first 

statement  of,  107 
second  statement  of,  108 


Wien's   law   of    spectral    distribu- 
tion, 108 

of   wave-length   of  maxi- 
mum energy,  108 
Work  done  by  electromotive  force, 

139 

by      magnetic      pole      in 
threading  a  circuit,  143 
by  evaporation,  84 
in  displacement  of  electrified 

system,  118 
performed    when    two    gases 

mix,  82 
Wrench  and  screw,  principle  of,  9 

Young's  modulus,  19 

Zeeman-effect,  187 
Zero  of  absolute  temperature,  defi- 
nition of,  63 
Zone-plate,  189 


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ym 

DEC  20  193 

OCT  281 

MAY  21  1934 


OCT  12  1938 


NOV    11  1942 
4 


NOV  26  1946 

!40ct'49RGC 
40ct'49J  B 


YC   11478 


UNIVERSITY  OF  CALIFORNIA  UBRARY 


